Determinants
Post by Deckmaster
I don't know about you, but when I was introduced to the concept of determinants, it was not easy or fun. We were just told "this is the determinant" and given a crazy formula. Okay, fine, so it wasn't that crazy, but it still wasn't nice and clean. Furthermore, it was not clear how to extend that 2-dimensional formula to 3-dimensional matrices, or beyond. I mean, I could have looked on Wikipedia to find a general formula, but that's actually a crazy formula. Furthermore, it doesn't actually explain what it's doing.
Calc 3 answered my cries. And now, I shall answer yours.
To start, let's look at a 3x3 matrix, A.
To find the determinant of A, we will break up A into sub-matrices. What we do is we take each element in the top row and multiply that by the determinant of the sub-matrix in A which doesn't contain the row or column of that element. Also, every other term in this sum will be negative.
"Woah, woah, what?" Yeah, pictures speak louder than words. Our determinant of A becomes:
See how we came up with that? We took the elements in green and multiplied by the determinant of the matrix of values in white, for each term:
"But we still need to take the determinants of matrices!" you cry. But the beauty of this algorithm is that it is recursive. If you know off the top of your head that the determinant of the first matrix is vz - wy, then great. If you don't, you just apply the algorithm again to get:
v * |z| - w * |y|. Since the determinant of a 1x1 matrix is the only element, you have your answer.
Now that you know this method, you can extend it to 4x4 matrices, 5x5 matrices, or any general nxn matrix for finding the determinant. Isn't this much easier to remember?
Calc 3 answered my cries. And now, I shall answer yours.
To start, let's look at a 3x3 matrix, A.
To find the determinant of A, we will break up A into sub-matrices. What we do is we take each element in the top row and multiply that by the determinant of the sub-matrix in A which doesn't contain the row or column of that element. Also, every other term in this sum will be negative.
"Woah, woah, what?" Yeah, pictures speak louder than words. Our determinant of A becomes:
See how we came up with that? We took the elements in green and multiplied by the determinant of the matrix of values in white, for each term:
"But we still need to take the determinants of matrices!" you cry. But the beauty of this algorithm is that it is recursive. If you know off the top of your head that the determinant of the first matrix is vz - wy, then great. If you don't, you just apply the algorithm again to get:
v * |z| - w * |y|. Since the determinant of a 1x1 matrix is the only element, you have your answer.
Now that you know this method, you can extend it to 4x4 matrices, 5x5 matrices, or any general nxn matrix for finding the determinant. Isn't this much easier to remember?
Comment by OmnipotentEntity on 9 Vigeo 10:4 - 17.56.97
Indeed it is, I was always shaky on what exactly determinants are used for though. Can you make a post on that?
Comment by Deckmaster on 9 Vigeo 10:4 - 17.62.1
I only know of one application as of now: vector cross products. If you want a post on that, I can do it at some point.
Comment by eofpi on 9 Vigeo 10:4 - 18.81.67
This is actually how I was taught determinants. The only determinant I was taught a formula for was the 2x2 case, i.e. vz-wy above. There's a faster way than this, though. I'll dig it up out of my calc 3 notes sometime.
Comment by Deckmaster on 9 Vigeo 10:5 - 16.13.45
Actually, I know of two uses. The second is in finding the Jacobian of a coordinate transformation.
eofpi would probably be better for writing an article on the application of this, though, he's higher in maths than I.
eofpi would probably be better for writing an article on the application of this, though, he's higher in maths than I.
Comment by Masqueradingasanerd22 on 9 Vigeo 11:0 - 11.93.80
i nvere herd uff deteeermninninz bfor toodae
Comment by Pi(e)3.14 on 9 Vigeo 12:0 - 14.74.72
Another way to think of it: a determinant is just the sum of all possible products of a number from the first row, a number from the second row in a different column, etc. with alternating signs based on which elements are chosen.
Can be used to find eigenvector/values and therefore create transformations. Also, a simple application: solving linear systems.
Can be used to find eigenvector/values and therefore create transformations. Also, a simple application: solving linear systems.
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