Question

find the determinant of each matrix. show work

-2,4,1
3,0,-1
1,2,1

Answer 1

I'll use a cofactor expansion along the second row:
\[\begin{align*}\begin{vmatrix}-2&4&1\\3&0&1\\1&2&1\end{vmatrix}&=(-3)\begin{vmatrix}4&1\\2&1\end{vmatrix}+0\begin{vmatrix}-2&1\\1&1\end{vmatrix}+(-1)\begin{vmatrix}-2&4\\1&2\end{vmatrix}\\
&=-3\begin{vmatrix}4&1\\2&1\end{vmatrix}-\begin{vmatrix}-2&4\\1&2\end{vmatrix}\\
&=-3(4-2)-(-4-4)\\
&=-6-(-8)\\
&=2
\end{align*}\]

Answer 2

wait how come you put a -3 up top when its a postitive?

Answer 3

Here's the pattern when doing a cofactor expansion:
\[\begin{bmatrix}+&-&+&\cdots\\
-&+&-&\cdots\\
+&-&+&\cdots\\
\vdots&\vdots&\vdots&\ddots\end{bmatrix}\]
Basically, what this means is that you either put a plus or minus sign on a given number, depending on where it is in the given matrix, and multiply that by the minor matrix that's left.