Inverting a 3x3 matrix using determinants Part 2: Adjugate matrix

we're nearing the homestretch of our quest to find the inverse of this and 3x3 matrix here and the next thing that we can do is find the determinant of it which we already have a good bit of practice doing so determinant the determinant of C of our matrix under the same color see there's several ways that you could do it several ways that you could do it you could say you could use the idea that it's equal to you could take this top row of the matrix and take the value of each of those terms times the cofactor times the corresponding cofactor and take the sum there that's one technique or you could do the technique where you rewrite these first two columns and then you take the product of the diagonals and end of the top two left diagonals sum those up and then subtract out the top the bottom the top right to the bottom left I'll do the second one just so that you can see that you get the same result so let's see the determinant is going to be equal to I'll rewrite all of these things so negative 1 negative 2 2 2 1 1 3 4 5 and let me now just to make it a little bit simpler rewrite these first two columns so negative 1 negative 2 2 1 3 4 so the determinant is going to be equal to so let me write this down so you have negative 1 times 1 times 5 well that's just going to be negative 5 taking that product then you have negative 2 times 1 times 3 well that's negative 6 so we'll have negative 6 or you can say plus negative 6 there and then you have 2 times 2 times 4 well that's just 4 times 4 which is just 16 so we have plus 16 and then we do the top right to the bottom left so you have negative 2 times 2 times 5 well that's negative 4 times 5 so that is negative 20 but we're going to subtract negative 20 so it's negative four times five negative twenty we're going to subtract negative 20 obviously that's going to turn into adding negative adding positive 20 then you have negative 1 times 1 times 4 which is negative 4 but we're going to subtract these products so this is we're going to subtract negative 4 and then you have 2 times 1 times 3 which is 6 but we have to subtract it so we have subtracting 6 and so this simplifies to negative 5 minus 6 is negative 11 negative 11 plus 16 gets us to positive 5 so all of this simplifies to positive 5 and then we have plus 20 plus 20 plus 4 so we do that green color so we don't get confused so we have plus 20 plus 4 plus 4 minus 6 so what does this get us 5 plus 20 is 25 plus 4 is 29 minus 6 29 minus 6 gets us to 23 so our determinant right over here is equal to is equal to 23 23 so now we are really in the homestretch the inverse of this matrix is going to be 1 over our determinant times the transpose of this cofactor matrix and the transpose of the cofactor matrix is called the adjutant so let's do that so let's write the ad you get here so C inverse this is drumroll we're really in the homestretch C inverse is equal to 1 over the determinant so it's equal to 1 over 23 times the adjutant of C and so this is going to be equal to 1 over 23 1 over 23 times the transpose of our cofactor matrix so we have our cofactor matrix right over here so each row now becomes a so this row this row now becomes a column so it becomes one negative seven five becomes the first column the second row becomes the second column 18 negative 11 negative 2 and then finally the third row becomes the third column you have negative 4 5 & 3 and now we just have to multiply or we could say divide each of these by 23 and we are there so this is going to be equal to this is the inverse of our original matrix C's homestretch 1/20 3 is just 1 21 20 thirds and you have 18 23rd actually let me give myself a little bit more real estate to do this in give myself a little bit more real estate so there we go so 1/20 3 123 18 18 23rd negative 4 23 negative 7 23 negative 11 23 5 20 thirds 523 negative 220 thirds and then finally assuming we haven't made any careless mistakes which would shock me if we haven't we get to 320 thirds and we are done we have successfully inverted a 3 by 3 matrix once again something I strongly believe better done by a computer and probably should not be part of a typical algebra 2 curriculum because it tend it tends to be displayed in a kind of non contextual way