Find the value of each determinant
1 answer:
Now, recall that, to get the determinant of a 3x3 or higher matrix, you pick a column or row to get the cofactors, also recall that the cofactors need to go witht the checkerboard pattern, so 
and so on.
so, let's use the last column, 2,0,4, because it has a zero there, and we can use that as a cofactor to simply get a 0 as a product with the minor.
![\bf \begin{bmatrix} 4&-1&\boxed{2}\\ 6&-1&\boxed{0}\\ 1&-3&\boxed{4} \end{bmatrix}\impliedby \textit{let's use the 3rd column for our cofactors}\\\\ -------------------------------\\\\ \begin{bmatrix} &&\boxed{2}\\ 6&-1&\\ 1&-3& \end{bmatrix}\implies +2 \begin{bmatrix} 6&-1\\1&-3 \end{bmatrix}\implies +2[(-18)-(-1)] \\\\\\ +2(-17)\implies -34\\\\ -------------------------------\\\\](https://tex.z-dn.net/?f=%5Cbf%20%5Cbegin%7Bbmatrix%7D%0A4%26-1%26%5Cboxed%7B2%7D%5C%5C%0A6%26-1%26%5Cboxed%7B0%7D%5C%5C%0A1%26-3%26%5Cboxed%7B4%7D%0A%5Cend%7Bbmatrix%7D%5Cimpliedby%20%5Ctextit%7Blet%27s%20use%20the%203rd%20column%20for%20our%20cofactors%7D%5C%5C%5C%5C%0A-------------------------------%5C%5C%5C%5C%0A%5Cbegin%7Bbmatrix%7D%0A%26%26%5Cboxed%7B2%7D%5C%5C%0A6%26-1%26%5C%5C%0A1%26-3%26%0A%5Cend%7Bbmatrix%7D%5Cimplies%20%2B2%0A%5Cbegin%7Bbmatrix%7D%0A6%26-1%5C%5C1%26-3%0A%5Cend%7Bbmatrix%7D%5Cimplies%20%2B2%5B%28-18%29-%28-1%29%5D%0A%5C%5C%5C%5C%5C%5C%0A%2B2%28-17%29%5Cimplies%20-34%5C%5C%5C%5C%0A-------------------------------%5C%5C%5C%5C)
![\bf \begin{bmatrix} 4&-1&\\ &&\boxed{0}\\ 1&-3& \end{bmatrix}\implies -0 \begin{bmatrix} 4&-1\\1&-3 \end{bmatrix}\implies -0[(-12)-(-1)]\implies 0\\\\ -------------------------------\\\\ \begin{bmatrix} 4&-1&\\ 6&-1&\\ &&\boxed{4} \end{bmatrix}\implies +4 \begin{bmatrix} 4&-1\\6&-1 \end{bmatrix}\implies +4[(-4)-(-6)] \\\\\\ +4[2]\implies 8\\\\ -------------------------------\\\\](https://tex.z-dn.net/?f=%5Cbf%20%5Cbegin%7Bbmatrix%7D%0A4%26-1%26%5C%5C%0A%26%26%5Cboxed%7B0%7D%5C%5C%0A1%26-3%26%0A%5Cend%7Bbmatrix%7D%5Cimplies%20-0%0A%5Cbegin%7Bbmatrix%7D%0A4%26-1%5C%5C1%26-3%0A%5Cend%7Bbmatrix%7D%5Cimplies%20-0%5B%28-12%29-%28-1%29%5D%5Cimplies%200%5C%5C%5C%5C%0A-------------------------------%5C%5C%5C%5C%0A%5Cbegin%7Bbmatrix%7D%0A4%26-1%26%5C%5C%0A6%26-1%26%5C%5C%0A%26%26%5Cboxed%7B4%7D%0A%5Cend%7Bbmatrix%7D%5Cimplies%20%2B4%0A%5Cbegin%7Bbmatrix%7D%0A4%26-1%5C%5C6%26-1%0A%5Cend%7Bbmatrix%7D%5Cimplies%20%2B4%5B%28-4%29-%28-6%29%5D%0A%5C%5C%5C%5C%5C%5C%0A%2B4%5B2%5D%5Cimplies%208%5C%5C%5C%5C%0A-------------------------------%5C%5C%5C%5C)
so our determinant comes down to the sum of those three products... so -34 -0 + 8.
and surely you know how much that is.
so, let's use the last column, 2,0,4, because it has a zero there, and we can use that as a cofactor to simply get a 0 as a product with the minor.
so our determinant comes down to the sum of those three products... so -34 -0 + 8.
and surely you know how much that is.
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