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There are probably many different ways to create your own typeface.I can share with you my 9-year learning experience in search for how to make the perfect handwriting font.It all began in 2007, when I had started an internship in the type department of FontShop International in Berlin. There, I learned about font technology, but didn’t yet know much about type design. That didn’t stop me from making my own handwriting font called Comic Jens. As you can tell by the name, it is a spoof on Comic Sans. I drew the letters with my mouse, just like Vincent Connare did with Comic Sans MS, using the brush tool in FontLab Studio 5. I didn’t worry about applying optical corrections and left everything as the brush tool produced it, doing only minimal cleanup of the outlines. I finished my work in 2008 and released the font for free. It was featured as an alternative to Comic Sans by some blogs and magazines and enjoyed moderate success. It was downloaded by more than 4000 people through the end of 2009.Now that I have more experience in type design, looking at the clumsy forms of the typeface makes me cringe. I can hardly stand to look at it anymore.But already back in 2009 I hatched the plan to rework the font completely and expand it into a large family of handwriting typefaces with lots of different weights and widths, something that hasn’t really been done for handwriting fonts at the same scale as for typographic superfamilies with over 100 styles like FF Clan by Łukasz Dziedzic or Thesis by Luc(as) de Groot. I intended to make around 10 weights and three widths (condensed, narrow, and normal), so about 30 fonts in total. I figured as my handwriting is already slightly slanted at about 3°, Italic styles would not be needed.I drew some test words without really having a plan how I could manage a project at this scale, and submitted a proposal to the FontShop type board, who accepted the design for inclusion in their FontFont typeface library. This encouraged me to keep working, but I still didn’t know how to pull this off.I spent a good part of the following year trying to figure out how I could turn my handwritten letters into digital outlines efficiently enough. FontShop required me to deliver at least a Latin Extended character set, which contains accented letters for all Latin-based European languages, plus four sets of numerals (lining proportional, lining tabular, oldstyle proportional, and oldstyle tabular), which amounts to something around 500 glyphs per font. Ideally, to keep up the illusion of handwritten letters, you have to include multiple variants of each glyph which are then automatically cycled through OpenType features as you type. At three glyph variants (the minimum, because three instances of the same letter can appear right next to each other in several languages), this would result in 1,500 glyphs per font.First I experimented with actually writing the letters using different sizes of markers to scan and trace them, but even before I scanned the first letter it became obvious to me that I just didn’t write well enough to use the letters as-is. I guess due to lack of training, plus missing calligraphy skills ... Another important lesson was that when you write and your letters work together well in the words you have written, doesn’t mean that they will work in different words and combinations.That is an important difference between lettering and type design. When designing a typeface, it is crucial to see the letters in context as you design them, to make sure the proportions harmonize (even in a handwriting typeface which allows for more inhomogeneity) and the letters work in different combinations. If you write a letter just a bit too large, you can’t just scale it down or the pen strokes will look too thin when compared to other letters. When you go for the look of a marker or felt tip pen, you end up with very messy traced contours made of lots of curve segments. You can’t really efficiently modify them in a font editor, especially when we are talking about 45,000 glyphs in total.My next try was to write the letters in Adobe Illustrator with a Wacom tablet. By using a pressure-sensitive brush tool I hoped to be able to keep the liveliness of written letters, as well as do some optical corrections right at this stage, like thinning the parts of letters where several strokes meet to avoid dark spots in the texture of the letters. Of course we don’t apply this kind of correction when we write on paper, but as I already wrote, designing a typeface is a whole different beast than writing letters.At least now I could put some reference letters next to the letter I was currently writing in my Illustrator template, and keep starting over with a letter until I was satisfied with the basic proportions in relation to my reference letters.The brush tool in Illustrator produces paths that consisted of a center line plus pressure (stroke expansion) information, but OpenType fonts are always made from outline contours. So I had to expand the paths before I could copy them over to FontLab Studio. It turned out the quality of the expanded paths was not that good that it wouldn’t require cleaning up (FontFont had very strict curve quality standards). Even if the expanded paths had been perfect, the prospect of repeating the whole process 45,000 times made me despair.I liked the clean look that writing on the tablet gave my letters. Most handwriting fonts keep some irregularities in the contours and texture of the letters, which looks fine at normal text sizes, but when zoomed in, or the letters being used very large, looks like crap because of the limited resolution of those features.My experience with Illustrator convinced me the only way to get the contour quality I wanted was to use the tablet-written letter only as a guide, and draft a completely new contour on top of it. That was probably still slow, but as I didn’t need exact written letters, but only a rough template for the proportions, I could ditch Illustrator and use FontLab Studio’s inferior brush tool. It didn’t support the tablet pressure information and produced even worse contours than Illustrator, but I just had to put them in the background layer and draw over them.Left: The contour as produced by FontLab’s brush toolRight: The manually redrawn contourWith a bit of practice and after getting up to speed, I managed to finish a glyph in 20 minutes from laying down the most important curve points on top of the template, to perfecting the curves and setting up the correct spacing of the glyph.Still I couldn’t see me doing this for 45,000 glyphs, in addition to working full-time in my regular job, too.I tried to write just the letters of the thinnest and the fattest weight and use a technique called interpolation, which is very common in today’s type design, to produce the intermediate weights. This requires an identical number of contours and the same control point structures in both ‘masters’ of each glyph, but since I was going to construct the contours manually this shouldn’t pose a problem. Or should it? When using interpolation, both masters of a glyph have to look roughly the same, or the interpolation will be possible mathematically, but not aesthetically pleasing. So this approach meant that when mixing different fonts of my family, the letters would look very similar in all weights and styles, not like handwriting at all.Again I was at a point where the whole project didn’t seem feasible. I wanted to finish something at least, so I decided to just draw a Regular and a Bold font independently, and just leave it at that. There went my plans for making the biggest handwriting font family in history.So I concentrated on the Regular weight first, and ambitiously planned to include a set of icons and symbols which are typical for comic fonts, like $#* used in place of expletives, arrows, moon phase symbols, animals, etc.I tried to keep the look of my letters formal, but not calligraphic. To me, my education in West Germany in the 1980s is reflected very much in these letter forms. I learned to write the ‘Vereinfachte Ausgangsschrift’ in primary school and I think it shows. To make the font more versatile, I consulted with an expert in the handwriting styles of different countries, Florian Hardwig, to include local alternate forms for some letters so users could tune the look of the text to match their local culture. I also added some forms of capital letters that were more closely based on the Roman capitalis monumentalis, to give all-caps settings a more classical look if desired (I’m straying quite far from comic lettering by now, you see).My colleague Christoph Koeberlin introduced me to the Pareto principle which says that you can achieve 80% of a result in 20% of the total time, and the remaining 20% will take 80% of the total time. And the nice thing in type design is that you can get away with cheating a lot. So my plan to include three variants of each glyph got downsized considerably.My guess was you didn’t really need three variants of every glyph, but just of the most common ones, like letters and numbers. Most handwriting fonts don’t include variants of letters at all, and hardly anybody notices. For good measure, I added a fourth variant of the twelve most frequently used letters of the English language: e t a o i n s h r d l u. The accented letters were built from parts, and not drawn separately, so they also had three or four variants.The finished Regular font has 1,454 glyphs. The Bold font does not contain all the symbols, and fewer letter variants, because it is only used for emphasis. These two fonts work very well in small sizes up to 12 points.I had finished both fonts when a lucky coincidence happened. A schoolbook publishing house contacted me because they were interested in using my ugly old free handwriting font, but required some design changes and more weights. I proposed to draw a new font family for them, similar in style to the old and new Comic Jens fonts, but more toned down to make it viable for use in text books and other school materials. They accepted my proposal, and we also agreed that I could release the fonts commercially later.So I started again from scratch with all the experience and routine I had gotten through the two finished fonts. Again I wrote on my tablet in FontLab Studio, one design with Hairline weight (20 units for the horizontal strokes on a 2000 units per em grid) and a Black weight (264 units for the horizontals). I had doubled the grid from the usual 1000 units per em because it was impossible to model the round stroke endings in a 20 unit line on a lower resolution. I wanted to interpolate the three requested weights (Light, Regular, Bold) between these extreme designs to be able to adjust the final stroke weights as needed. Imagine drawing a Light weight by hand and then the client decides it should be a tad lighter ... you would have to adjust everything manually again.The new letters were more regularized in appearance for school use, and I had the idea that they could also make a nice headline-size addition to my two text-size fonts. When you write larger, you write more slowly, and your letters don’t jump around as much as when you write small. The publisher wanted their fonts even more toned down, so in the end I did different versions of a number of letters for their selection of weights and for my headline fonts. For my fonts, I decided to produce eleven weights: Hairline, Thin, UltraLight, ExtraLight, Light, Regular, Book, Medium, Bold, ExtraBold and Black.I had switched from FontLab Studio 5 to RoboFont as my main font editor by now. RoboFont is much more customizable. The whole interpolation process was a controlled by custom Python scripts to guarantee reproducible results for both branches of the design. All files were kept in a git repository for version and branch control.The interpolation of the two extreme fonts proved helpful, but it was not possible to use only two masters for interpolation. In the Black master, the stroke weights are very variable, bold in wide curves but thinned out where strokes join, while the Hairline master is almost monolinear. When you interpolate a font halfway between these, you end up with a Regular weight where the stroke joins are way too thin. Some letters had to be designed differently in different weights, such as the dollar sign or the sterling sign, where the loop closes up at a certain weight.The round stroke endings also proved difficult to interpolate. The whole font is drawn with virtually no straight lines, just curves. If you interpolate two Bézier curves, the angles at the start and end point of the curve will change if they are not exactly the same angle in the two masters, or if the distance of the control points to the curve point keep exactly the same ratio. You see, there is a lot of math involved at some point.With the large difference of weight and the irregularity of the design between the Hairline and Black masters, the exact conditions for smooth curve points were impossible to maintain. So the process was to interpolate a medium weight, do the optical correction of thickening the stroke joins manually, and to check if all the smooth connections between curves really stayed smooth. They didn’t, and I wrote another Python script that would analyze the curves and automatically adjust most of the kinks that had crept in. Still I had to visually check every one of the 733 glyphs (the Headline styles also have a smaller glyph set than the Text styles). Also, the script smoothed out some corners that actually were there by design, which I had to check and undo on a case by case basis.Green is the raw interpolated version, black the corrected versionAfter inserting the third master into the interpolation space, the stroke weights in the remaining 8 interpolated weights looked good, but my script and visual check to correct the curve transitions were still necessary and took quite a while.After this, my job was done. Until the fonts are ready for release, there is still a lot more work to do, which I happily delegated to Inka Strotmann at Monotype (FontShop had been bought by Monotype in the meantime). In addition to doing the screen optimization, naming, final font production in OTF, TTF and WOFF formats, and quality assurance, most importantly she looked at my designs with a pair of fresh eyes and gave me some suggestions for changes, and found some errors that are inevitable because you just don’t notice them anymore when you spend such a long time looking at and working with your letters.A bit of a letdown late in the process was that the lawyers were uncomfortable with the name FF Comic Jens, because they feared that someone might consider it infringing on the Comic Sans MS trademark, though the name Comic Jens had been out there since 2008 and nobody took offence. I finally found a new name that I’m happy with: FF Uberhand. It is the uber handwriting font family, the handwriting font to rule all other handwriting fonts, and also in German ‘überhand nehmen’ means that something gets out of hand ;)I’m happy to say that the fonts are scheduled for release later this year.The FF Uberhand family in all its gloryIn-use example for a Valentine’s Day card

Capital Letters ( according to Sussex University)Capital letters are not really an aspect of punctuation, but it is convenient to deal with them here. The rules for using them are mostly very simple.(a) The first word of a sentence, or of a fragment, begins with a capital letter:The bumbling wizard Rincewind is Pratchett's most popular character.Will anyone now alive live to see a colony on the moon? Probably not.Distressingly few pupils can locate Iraq or Japan on a map of the world.(b) The names of the days of the week, and of the months of the year, are written with a capital letter:Next Sunday France will hold a general election.Mozart was born on 27 January, 1756.Football practice takes place on Wednesdays and Fridays.However, the names of seasons are not written with a capital:Like cricket, baseball is played in the summer.Do not write *"... in the Summer".(c) The names of languages are always written with a capital letter. Be careful about this; it's a very common mistake.Juliet speaks English, French, Italian and Portuguese.I need to work on my Spanish irregular verbs.Among the major languages of India are Hindi, Gujarati and Tamil.These days, few students study Latin and Greek.Note, however, that names of disciplines and school subjects are not capitalized unless they happen to be the names of languages:I'm doing A-levels in history, geography and English.Newton made important contributions to physics and mathematics.She is studying French literature.(d) Words that express a connection with a particular place must be capitalized when they have their literal meanings. So, for example, French must be capitalized when it means `having to do with France':The result of the French election is still in doubt.The American and Russian negotiators are close to agreement.There are no mountains in the Dutch landscape.She has a dry Mancunian sense of humour.(The word Mancunian means `from Manchester'.)However, it is not necessary to capitalize these words when they occur as parts of fixed phrases and don't express any direct connection with the relevant places:Please buy some danish pastries.In warm weather, we keep our french windows open.I prefer russian dressing on my salad.Why the difference? Well, a danish pastry is merely a particular sort of pastry; it doesn't have to come from Denmark. Likewise, french windows are merely a particular kind of window, and russian dressing is just a particular variety of salad dressing. Even in these cases, you can capitalize these words if you want to, as long as you are consistent about it. But notice how convenient it can be to make the difference:In warm weather, we keep our french windows open.After nightfall, French windows are always shuttered.In the first example, french windows just refers to a kind of window; in the second, French windows refers specifically to windows in France.(e) In the same vein, words that identify nationalities or ethnic groups must be capitalized:The Basques and the Catalans spent decades struggling for autonomy.The Serbs and the Croats have become bitter enemies.Norway's most popular singer is a Sami from Lapland.(An aside: some ethnic labels which were formerly widely used are now regarded by many people as offensive and have been replaced by other labels. Thus, careful writers use Black, not Negro; native American, not Indian or red Indian; native Australian, not Aborigine. You are advised to follow suit.)(f) Formerly, the words black and white, when applied to human beings, were never capitalized. Nowadays, however, many people prefer to capitalize them because they regard these words as ethnic labels comparable to Chinese or Indian:The Rodney King case infuriated many Black Americans.You may capitalize these words or not, as you prefer, but be consistent.(g) Proper names are always capitalized. A proper name is a name or a title that refers to an individual person, an individual place, an individual institution or an individual event. Here are some examples:The study of language was revolutionized by Noam Chomsky.The Golden Gate Bridge towers above San Francisco Bay.There will be a debate between Professor Lacey and Doctor Davis.The Queen will address the House of Commons today.Many people mistakenly believe that Mexico is in South America.My friend Julie is training for the Winter Olympics.Next week President Clinton will be meeting Chancellor Kohl.Observe the difference between the next two examples:We have asked for a meeting with the President.I would like to be the president of a big company.In the first, the title the President is capitalized because it is a title referring to a specific person; in the second, there is no capital, because the word president does not refer to anyone in particular. (Compare We have asked for a meeting with President Wilson and *I would like to be President Wilson of a big company.) The same difference is made with some other words: we write the Government and Parliament when we are referring to a particular government or a particular parliament, but we write government and parliament when we are using the words generically. And note also the following example:The patron saint of carpenters is Saint Joseph.Here Saint Joseph is a name, but patron saint is not and gets no capital.There is a slight problem with the names of hazily defined geographical regions. We usually write the Middle East and Southeast Asia, because these regions are now regarded as having a distinctive identity, but we write central Europe and southeast London, because these regions are not thought of as having the same kind of identity. Note, too, the difference between South Africa (the name of a particular country) and southern Africa (a vaguely defined region). All I can suggest here is that you read a good newspaper and keep your eyes open.Observe that certain surnames of foreign origin contain little words that are often not capitalized, such as de, du, da, von and van. Thus we write Leonardo da Vinci, Ludwig van Beethoven, General von Moltke and Simone de Beauvoir. On the other hand, we write Daphne Du Maurier and Dick Van Dyke, because those are the forms preferred by the owners of the names. When in doubt, check the spelling in a good reference book.A few people eccentrically prefer to write their names with no capital letters at all, such as the poet e. e. cummings and the singer k. d. lang. These strange usages should be respected.(h) The names of distinctive historical periods are capitalized:London was a prosperous city during the Middle Ages.Britain was the first country to profit from the Industrial Revolution.The Greeks were already in Greece during the Bronze Age.(i) The names of festivals and holy days are capitalized:We have long breaks at Christmas and Easter.During Ramadan, one may not eat before sundown.The feast of Purim is an occasion for merrymaking.Our church observes the Sabbath very strictly.The children greatly enjoy Hallowe'en.(j) Many religious terms are capitalized, including the names of religions and of their followers, the names or titles of divine beings, the titles of certain important figures, the names of important events and the names of sacred books:An atheist is a person who does not believe in God.The principal religions of Japan are Shinto and Buddhism.The Indian cricket team includes Hindus, Muslims, Sikhs and Parsees.The Lord is my shepherd.The Prophet was born in Mecca.The Last Supper took place on the night before the Crucifixion.The Old Testament begins with Genesis.Note, however, that the word god is not capitalized when it refers to a pagan deity:Poseidon was the Greek god of the sea.(k) In the title or name of a book, a play, a poem, a film, a magazine, a newspaper or a piece of music, a capital letter is used for the first word and for every significant word (that is, a little word like the, of, and or in is not capitalized unless it is the first word):I was terrified by The Silence of the Lambs.The Round Tower was written by Catherine Cookson.Bach's most famous organ piece is the Toccata and Fugue in D Minor.I don't usually like Cher, but I do enjoy The Shoop Shoop Song.Important note: The policy just described is the one most widely used in the English-speaking world. There is, however, a second policy, preferred by many people. In this second policy, we capitalize only the first word of a title and any words which intrinsically require capitals for independent reasons. Using the second policy, my examples would look like this:I was terrified by The silence of the lambs.The round tower was written by Catherine Cookson.Bach's most famous organ piece is the Toccata and fugue in D minor.I don't usually like Cher, but I do enjoy The shoop shoop song.You may use whichever policy you prefer, so long as you are consistent about it. You may find, however, that your tutor or your editor insists upon one or the other. The second policy is particularly common (though not universal) in academic circles, and is usual among librarians; elsewhere, the first policy is almost always preferred.(l) The first word of a direct quotation, repeating someone else's exact words, is always capitalized if the quotation is a complete sentence:Thomas Edison famously observed "Genius is one per cent inspiration and ninety-nine per cent perspiration."But there is no capital letter if the quotation is not a complete sentence:The Minister described the latest unemployment figures as "disappointing".(m) The brand names of manufacturers and their products are capitalized:Maxine has bought a second-hand Ford Escort.Almost everybody owns a Sony Walkman.Note: There is a problem with brand names which have become so successful that they are used in ordinary speech as generic labels for classes of products. The manufacturers of Kleenex and Sellotape are exasperated to find people using kleenex and sellotape as ordinary words for facial tissues or sticky tape of any kind, and some such manufacturers may actually take legal action against this practice. If you are writing for publication, you need to be careful about this, and it is best to capitalize such words if you use them. However, when brand names are converted into verbs, no capital letter is used: we write She was hoovering the carpet and I need to xerox this report, even though the manufacturers of Hoover vacuum cleaners and Xerox photocopiers don 't much like this practice, either.(n) Roman numerals are usually capitalized:It is no easy task to multiply LIX by XXIV using Roman numerals.King Alfonso XIII handed over power to General Primo de Rivera.The only common exception is that small Roman numerals are used to number the pages of the front matter in books; look at almost any book.(o) The pronoun I is always capitalized:She thought I'd borrowed her keys, but I hadn't.It is possible to write an entire word or phrase in capital letters in order to emphasize it:There is ABSOLUTELY NO EVIDENCE to support this conjecture.On the whole, though, it is preferable to express emphasis, not with capital letters, but with italics. It is not necessary to capitalize a word merely because there is only one thing it can possibly refer to:The equator runs through the middle of Brazil.Admiral Peary was the first person to fly over the north pole.The universe is thought to be about 15 billion years old.Here the words equator, north pole and universe need no capitals, because they aren't strictly proper names. Some people choose to capitalize them anyway; this is not wrong, but it's not recommended.Capital letters are also used in writing certain abbreviations and related types of words, including the abbreviated names of organizations and companies, and in letter writing and in the headings of essays.There is one other rather rare use of capital letters which is worth explaining if only to prevent you from doing it by mistake when you don't mean to. This to poke fun at something. Here is an example:The French Revolution was a Good Thing at first, but Napoleon's rise to power was a Bad Thing.Here the writer is making fun of the common tendency to see historical events in simple-minded terms as either good or bad. Another example:Many people claim that rock music is Serious Art, deserving of Serious Critical Attention.The writer is clearly being sarcastic: all those unusual capital letters demonstrate that he considers rock music to be worthless trash.This stylistic device is only appropriate in writing which is intended to be humorous, or at least light-hearted; it is quite out of place in formal writing.The use of unnecessary capital letters when you're trying to be serious can quickly make your prose look idiotic, rather like those content-free books that fill the shelves of the "New Age" section in bookshops:Your Eidetic Soul is linked by its Crystal Cord to the Seventh Circle of the Astral Plane, from where the Immanent Essence is transmitted to your Eidetic Aura,...You get the idea. Don't use a capital letter unless you're sure you know why it's there.Summary of Capital Letters:Capitalizethe first word of a sentence or fragmentthe name of a day or a monththe name of a languagea word expressing a connection with a placethe name of a nationality or an ethnic groupa proper namethe name of a historical periodthe name of a holidaya significant religious termthe first word, and each significant word, of a titlethe first word of a direct quotation which is a sentencea brand namea Roman numeralthe pronoun I

Topic: linear algebra/ Level: freshman/ About: determinantsLet’s start with most basic concepts first. A determinant is a property of a matrix.Just so we’re on the same page, let’s see what a Matrix really is. I could type something like this in Julia language:[1 3 3; 4 7 6 ] And get this matrix as a result:2×3 Array{Int64,2}:  1 3 3  4 7 6 Julia informs us that we have a 2-row-by-3-column “array” (a fancy name for a matrix), which is made up of numbers (Julia calls them double-precision integers: 64 refers to double precision, meaning the numbers will not easily run into various errors when you’re computing. You can disregard this bit.)So what have we done so far? We have represented a linear equation in a number-box form. The equation corresponding to this matrix is:[math]x+3y+3t [/math][math]4x+7y+6t[/math]Note that we said equation. But the above two expressions do not have an equation. This is because they are in coefficient form: this is saying, we have intentionally removed the right side of the equation.Suppose we want a complete matrix—what is called augmented matrix. Then we would have this equation:[math]x+3y+3t = 90[/math][math]4x+7y+6t = 138[/math](90 and 138 are also arbitrarily chosen by me, and they can be any number you like).However, when we represent this linear equation as a matrix, we cannot place the 90 and 138 in the same space as the original matrix. So the coefficient matrix is made up of only the coefficient numbers of the unknown variables—in our cases the variables are x, y, t (but they can go by any name, including x1, x2, x3 etc).So let’s call our matrix a name. Any capital letter would do, so I choose S.Now, we can represent the above equation as [S | b], where S stands for the coefficient matrix, and b stands for the right-side of the equation numbers.Recall also that any group of ordered and same-type elements can be called a vector. That is, if I put a few numbers together, in a specific order, I can call them a vector.So we can see that b here is a vector, and in Julia we represent it so:b = [90 138] In this case, b is a vector in [math]R^2. [/math]This is saying b is made up of real numbers, and has only 2 numbers in it. The quantity of numbers refers to how this vector would appear, had we plotted it. In the case of b, we have only two directions, x and y: x is 90, y is 138.So if we were to plot b, we could start from the Cartesian plane’s origin (just a big plus sign on a paper, the origin being the intersection of the vertical and horizontal lines), and as x is the horizontal one, we go 90 steps to the right (since right is positive on the number line) and 138 steps up (up is positive).If now, from where we stand, we drew a straight and diagonal line to the origin, this would be a geometric representation of our vector b. So where does a determinant come into play here? We’re getting there, patience!The reason we converted our SLE, system of linear equations, into a box of numbers, a matrix, is because we want to solve the SLE and reduce visual and computational clutter. Also, in our case, the system had two equations, or two rows. (Note that it is called a system because the solution values for x, y and t should be the same for all rows).Matrices come in two main shapes: square and rectangular. The square ones are also also called [math]n x n[/math], meaning, the number of rows is equal to the number of columns, whereas the rectangular ones are generally called [math]m x n[/math] where [math]m[/math] stands for rows and [math]n[/math] for columns.Of course, when we represent matrices in these square and rectangular dimensional ways ([math]n x n[/math] and [math]m x n[/math]) they usually mean the coefficient matrix. Also please note that when the matrix is not in coefficient form, it’s a good practice to mention that the matrix in question is augmented, or represents the whole SLE).Our matrix S, therefore, is rectangular.For the purpose of this question, we won’t have anything to do with rectangular matrices, since, point 1: determinants are only for square matrices.So let’s produce a random square matrix and call it H:H = [2 4 ; 8 16]  # output  > 2×2 Array{Int64,2}:  2 4  8 16 So now we can ask two questions: one is, what is the solution to H, if its b vector is [80, 120]? But we can also save time and simply ask, does H have any solution at all?The second question is useful in practice because sometimes the coefficient matrix may be the same, but the b vector may continue to change. So suppose that we are adamant that we need to know the solution.So if we try this: [H |b1] and don’t get a solution, we are compelled to say two things: [H |b1] has no solution, but [H |b2] may have. So that will cause us to try solving [H |b2], and if we don’t have a solution, then continue wasting time in this manner for all [H | bn].On the contrary, if we simply find out whether H has a solution, then our problem is mostly solved. Why? Because if H has no solution, we discard it. Time saved. If it has solution, we can then proceed to solve it, and there, if we get an error or no solution, we will have known that our computation had an error and not that H had a problem.So point 2: if the determinant of a square matrix is not zero, it definitely has a solution.To find a square matrix’s determinant by hand, we can only solve problems that involve matrices in 2 or 3 dimensions. Higher dimensions cannot be solved by hand (too much work and error-prone), so we solve them with Julia, like so:using LinearAlgebra det(H)  > 0.0 We have imported a Julia library (code package) called Linear Algebra, so that we can use the det() function—instead of handcoding it ourselves.If you are curious, this is the mathematical procedure for finding the determinant of a 2x2 matrix we call M:[math]M = \left (\begin{array}{cc} w & z \\ k & o \end{array}\right )[/math]The determinant is simply [math]wo-zk[/math].Why? because a determinant not only determines a solution for a matrix, it also helps us find it (if we wanted to go that way; there are faster ways to find the solution). Given this SLE:[math]\begin{align} wx + zy & = b \\ kx + oy & = t \end{align}[/math]The solution can be had as:[math]x = \frac{ob - zt}{wo - zk}, y = \frac{wz - kb}{wo - zk}.[/math]Now that we know the “historical” significance of a determinant, I want to hasten to add that this is not the only thing it is good for. Point 3: with regard to inverse matrices, if the determinant of a square matrix is zero, the matrix does not have an inverse.Just as 5 is inverse of 1/5, so a square matrix can have an inverse version; note that since matrices do not divide (1/M or M/1 are both incorrect), the inverse of a matrix is something like this:[math]M[/math] and [math]M^{-1}[/math] are said to be inverses of each other, if their product gives us the identity matrix, generally called [math]I[/math].The identity matrix is simply a square matrix that mostly filled with zeros, except that its left diagonal (the end starting from w and going to o in our 2x2 example) is all 1’s.Here is a 2x2 identity matrix:1 0  0 1 Here is a 4x4 identity matrix. Note the left diagonal (also called Main diagonal is all 1’s):1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 So what is an inverse of a matrix for? This question is relevant since we need the determinant to identify invertible matrices—hence, we need to know the benefits of invertible matrices.Quite simply, since a non-zero determinant of a matrix decides the existence of a solution (which is always unique, if it exists) for a given square matrix, and since a non-zero determinant also indicates that a matrix is invertible, thus all invertible matrices have a unique solution.Consider our earlier augmented matrix: [H |b1]. We established earlier that its determinant is zero.To solve the SLE that is represented by [H |b1], we can simply use the Gauss-Jordan elimination method (digression, so I am not going into that rabbit hole), or use the inverse method to solve. To do so, we represent it in a slightly different way:Hx = b1Note that all small letter symbols in bold stand for vectors and capital letters are reserved for matrices. Now then, Hx = b1 and [H |b1] are representing the same SLE; the only difference is that we took the unknown variables and represented them as a vector like so: 2t 4y  8t 16y  H Becoming:2 4  8 16 and the t and y are both represented as a single vector:x = [t, y] So to solve this, we simply do algebraic reordering:[math]x = H^{-1}*b1[/math]You notice that as soon as H moves to the other side of the equation, it becomes inverse.Do we have to do this? Luckily, no, since H’s determinant is zero, so doing this operation is pointless. That is, H has no solution. But if a matrix has a non-zero determinant, it is invertible, and therefore lends itself to such algebraic reordering.Inverse matrices have many other uses; for instance, they can be used to simplify or prove problems involving Matrices. Consider this equation:[math](AB)^{-1} = B^{-1}A^{-1}[/math]Is it true? Intuition says no, since a Matrix does not behave like a real number (that is AB is not the same as BA as we saw earlier) . So now we must prove or disprove this.Let’s make use of the properties of the inverse matrix here. Multiply both sides to AB:[math]AB * (AB)^{-1} = AB * B^{-1}A^{-1} [/math]The left side, according to what we saw earlier turns into an identity matrix:[math]M * M^{-1} = I[/math] (This is the same as [math]AB * (AB)^{-1})[/math]Now, on the right side the middle matrices [math]B^{-1} * B[/math] will evaluate to identity:[math]AB * B^{-1}A^{-1} = A*I*A^{-1}[/math]We also know a matrix multiplied by [math]I[/math] is equal to itself:[math]M * I = M[/math]Thus:[math]A*I*A^{-1} = A * A^{-1} = I[/math]This also evaluates to identity, proving our original equation (since both sides evaluated to an identity matrix).Finally, the determinant has many other advanced uses which I won’t mention because they are outside the scope of this introductory answer.All the best,Jan