Answer to Question #196775 in Linear Algebra for Mapula Advice

Question

Compute all the minors is and cofactors of (123 201 234)

Answer 1

Given matrix is-

$A=\begin{bmatrix} 1&2&3\\2&0&1\\2&3&4\end{bmatrix}$

Minors are

$M_{11}=\begin{vmatrix} 0&1\\3&4\end{vmatrix}=-3\\[9pt]M_{12}=\begin{vmatrix} 2&1\\2&4\end{vmatrix}=6\\[9pt]M_{13}=\begin{vmatrix} 2&0\\2&3\end{vmatrix}=6\\[9pt]M_{21}=\begin{vmatrix} 2&3\\3&4\end{vmatrix}=-1\\[9pt]M_{22}=\begin{vmatrix} 1&3\\2&4\end{vmatrix}=-2\\[9pt]M_{23}=\begin{vmatrix} 1&2\\2&3\end{vmatrix}=-1\\[9pt]M_{31}=\begin{vmatrix} 2&3\\0&1\end{vmatrix}=2\\[9pt]M_{32}=\begin{vmatrix} 1&3\\2&1\end{vmatrix}=-5\\[9pt]M_{33}=\begin{vmatrix} 1&2\\2&0\end{vmatrix}=-4\\$

Cofactor is-

$C_{ij}=(-1)^{i+j}M_{ij}$

=$\begin{bmatrix} -3&-6&6\\1&-2&1\\2&5&-4\end{bmatrix}$