Question

Comupute all the minors and the cofactors of: [1 2 3]

[2 0 1]

[2 3 4]

Answer 1

Solution.

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

The elements of the minor of the matrix A will be given by

$M_{1,1}=\begin{vmatrix} 0 & 1\\ 3& 4 \end{vmatrix}=-3$

$M_{1,2}=\begin{vmatrix} 2& 1\\ 2& 4 \end{vmatrix}=6$

$M_{1,3}=\begin{vmatrix} 2 & 0\\ 2& 3 \end{vmatrix}=6$

$M_{2,1}=\begin{vmatrix} 2& 3\\ 3& 4 \end{vmatrix}=-1$

$M_{2,2}=\begin{vmatrix} 1 & 3\\ 2& 4 \end{vmatrix}=-2$

$M_{2,3}=\begin{vmatrix} 1 & 2\\ 2& 3 \end{vmatrix}=-1$

$M_{3,1}=\begin{vmatrix} 2 & 3\\ 0& 1 \end{vmatrix}=2$

$M_{3,2}=\begin{vmatrix} 1& 3\\ 2& 1 \end{vmatrix}=-5$

$M_{3,3}=\begin{vmatrix} 1& 2\\ 2& 0 \end{vmatrix}=-4$

Hence, $M=\begin{pmatrix} -3& 6&6 \\ -1& -2&-1\\ 2&-5&-4 \end{pmatrix}$

The cofactor matrix will be given by $C_{i,j}=(-1)^{i+j}M_{i,j}.$

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

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