This topic could be placed in a number of mathematical categories (numbers, linear algebra, matrices, calculus, and others).
I toyed with listing it in the [Algebra] section of this website, because determinants were first created as a method for rapidly solving simultaneous linear equations. Once mathematicans realised that they were really numbers, however, they became curious about their properties … how to add, subtract, multiply and divide them (and more). So, that is why you find them here, among the numbers.
You can see, from the illustration to the left, that the determinant of the 2×2 matrix (represented with parentheses) is written using the same array of numbers with a vertical line on either side. For a 2×2 matrix, the determinant is calculated (evaluated) by multiplying the numbers on the leading diagonal (3)(7) and subtracting the product of the remaining numbers on the other diagonal (5)(-1). As you can see, this results in a single number.
Why are they used, what are they useful for, and how they are manipulated will be revealed on this page in time. I will also be sharing about the history of linear equations, matrices and determinants.
I was going to provide a link to a useful site for you to use in the meantime, but I cannot find a site that starts from the basic properties of an array of linear equations. I believe in starting with such an array and then deducing the concepts of matrix multiplication, inverse matrices, transposed matrices, and determinants. Please be patient.
I have been watching your “Integration technique” videos with high expectation and excitement. They never fail to give me a lot of joy and much pleasure. You must have been one hell of teacher at your time! … just continue making highly enjoyable and intellectually stimulating videos. … Unfortunately, what you learn at the university, you hardly ever see again in your office job. That is why I try to repeat some thing I learned: to enjoy it a bit more, to marvel upon beauty, and to keep my brain sharp. There is where your videos come in perfectly: interesting, amusing and thought provoking problems neatly presented. … You do your best to share your passion, knowledge and experience. The least I can do is to say an honest “Thank you!” As I said, your videos give me much pleasure. You may freely let other people know that. … Rest assured that I shall be watching your videos in the future. With joy and excitement. And I shall comment on them from time to time. You just continue making them.
Zoran V (in private correspondence via YouTube, quoted with permission)
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