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 [tex]\bf \begin{bmatrix}
+&-&+&-&+&...\\
-&+&-&+&...\\
+&-&+&-&+&...
\end{bmatrix}[/tex] 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.
[tex]\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\\\\
-------------------------------\\\\[/tex]
[tex]\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\\\\
-------------------------------\\\\[/tex]
so our determinant comes down to the sum of those three products... so -34 -0 + 8.
and surely you know how much that is.
