Determinant of a 3x3 matrix: shortcut method (2 of 2)
About this video
Sal finds the determinant of the same 3x3 matrix A again, this time with the shortcut of rewriting the first two columns to the right. He adds the products of the three top-left to bottom-right diagonals and subtracts the products of the three top-right to bottom-left diagonals. The result, 6 + 10 = 16, matches the standard method.
You'll be able to
- set up the shortcut method by rewriting the first two columns.
- add the down-right diagonal products and subtract the down-left diagonal products.
- use the shortcut to find the determinant of a 3x3 matrix.
Key words
- determinant
- A single number calculated from a square matrix.
- diagonal product
- The product of three entries lying along one diagonal line in the extended grid.
Transcript
As another hint, I will take the same matrix, matrix A and take its determinant again but I will do it using a different technique, either technique is valid so here we saying what is the determinant of the 3X3 Matrix A and we can is we can rewrite first two column so first column right over here we could rewrite it as 4 4 -2 and then the second column right over here we could rewrite it -1 5 0 and we could do is we could take the sum of the products of the first three top left bottom left diagonal, let me show you so the product of that that plus that plus that trying my best to draw this neatly and then from that subtract top right to bottom left diagonal so from that subtract let get a color
I havent used subtract that and that and that sounds really confusing with all the diagonals i have drawn lets look at the blue ones first 4 times 5 times 0 4 times 5 times 0 plus -1 times 3 times -2 plus -1 times 3 times -2 let me put these in parenthesis plus 1 times 4 times 0 and then we gonna subtract all of these orange diagonal, we go from the top right to the bottom left so we gonna subtract we could do 1 times 5 times -2 1 times 5 times -2 and then we can subtract subtract 4 times 3 times 0 4 times 3 times 0 and then we can subtract -1 times 4 times 0 -1 times 4 times 0 and now we just evaluate this over here 4 times 5 times 0 is just 0 -1 times 3 times -2 is +6 so this is +6 1 times 4 times 0 is 0 once again
and then we have 1 times 5 times -2 is -10 but we have this negative out here so it becomes a +10 then 4 times 3 times , well thats just going to be 0. and then we have -1 times 4 times 0, which is just 0 so we are left with +6 + 10 which is equal to +16
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