## Related questions with answers

List the minor matrix $M_{ij}$, and calculate the cofactor $A_{i j}=(-1)^{i+j} \operatorname{det}\left(M_{i j}\right)$ for the matrix A given by

$A=\left[\begin{array}{rrrr} 2 & -1 & 3 & 1 \\ 4 & 1 & 3 & -1 \\ 6 & 2 & 4 & 1 \\ 2 & 2 & 0 & -2 \end{array}\right]$

$M_{11}$

Solution

VerifiedWe have

$\begin{align*} A&=\begin{bmatrix} 2&-1&3&1\\ 4&1&3&-1\\ 6&2&4&1\\ 2&2&0&-2 \end{bmatrix} \end{align*}$

Find $M_{11}$, according to $\text{\color{#c34632}Definition 3}$, the minor matrix $M_{11}$ is the determinant of the matrix $A$ formed by deleting the first row and the first column from the matrix $A$, then

$\begin{align*} M_{11}&=\begin{bmatrix} 1&3&-1\\ 2&4&1\\ 2&0&-2 \end{bmatrix} \end{align*}$

then find

$\begin{align*} A_{11}&=(-1)^{1+1}\det (M_{11})\\ &=\det \begin{vmatrix} 1&3&-1\\ 2&4&1\\ 2&0&-2 \end{vmatrix}\\ &=1\cdot \det \begin{vmatrix} 4&1\\ 0&-2 \end{vmatrix}-3\cdot \det \begin{vmatrix} 2&1\\ 2&-2 \end{vmatrix}-1\cdot \det \begin{vmatrix} 2&4\\ 2&0 \end{vmatrix}\\ &=1(-8-0)-3(-4-2)-1(0-8)\\ &=-8+18+8\\ &=18 \end{align*}$

Hence, $M_{11}=18$.

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