Answer to Question #196878 in Linear Algebra for Mpopo

Question

Compute all the minors and cofactors of

1 2 3

2 0 1

2 3 4

Answer 1

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

Let $M_{ij}$ stands for Minors and $C_{ij}$ is cofactors

where $i=i^{th}$ row and $j=j^{th }$ column of the matrix

$M_{11}=\begin{vmatrix} 0 & 1 \\ 3 & 4 \end{vmatrix}=0-3=-3\ \ \ ;\ \ C_{11}=(-1)^{1+1}M_{11}=-3\\\ \\M_{12}=\begin{vmatrix} 2 & 1 \\ 2 & 4 \end{vmatrix}=8-2=6\ \ ;\ \ C_{12}=(-1)^{1+2}M_{12}=-6\\\ \\M_{13}=\begin{vmatrix} 2 & 0 \\ 2 & 3 \end{vmatrix}=6-0=6\ \ ;\ \ C_{13}= (-1)^{1+3}M_{13}=6\\\ \\M_{21}=\begin{vmatrix} 2 & 3 \\ 3 & 4 \end{vmatrix}=8-9=-1\ \ ;\ \ C_{13}=(-1)^{2+1}M_{21}=1\\\ \\M_{22}=\begin{vmatrix} 1 & 3 \\ 2 & 4 \end{vmatrix}=4-6=-2\ \ ;\ \ C_{22}=(-1)^{2+2}M_{22}=-2\\\ \\M_{23}=\begin{vmatrix} 1 & 2 \\ 2 & 3 \end{vmatrix}=3-4=-1\ \ ;\ \ C_{23}=(-1)^{2+3}M_{23}=1\\\ \\M_{31}=\begin{vmatrix} 2 & 3 \\ 0 & 1 \end{vmatrix}=2-0=2\ \ ;\ \ C_{31}=(-1)^{3+1}M_{31}=2 \\\ \\M_{32=}\begin{vmatrix} 1 & 3 \\ 2 & 1 \end{vmatrix}=1-6=-5\ \ ;\ \ C_{32}=(-1)^{3+2}M_{32}=5\\\ \\M_{33}=\begin{vmatrix} 1 & 2 \\ 2 & 0 \end{vmatrix}=0-4=-4\ \ ;\ \ C_{33}=(-1)^{3+3}M_{33}=-4$