Find the determinants of minors and cofactors of the determinant $|\begin{matrix} 2 & 3 & 4 \\ 7 & 2 & - 5 \\ 8 & - 1 & 3 \end{matrix}|$

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$|\begin{matrix} 1 & - 61 & - 23 \\ - 13 & - 26 & 26 \\ - 23 & 38 & - 17 \end{matrix}|$ and $|\begin{matrix} 1 & 61 & - 23 \\ 13 & - 26 & - 26 \\ - 23 & - 38 & - 17 \end{matrix}|$

$|\begin{matrix} 1 & 61 & - 23 \\ 13 & - 26 & - 26 \\ - 23 & - 38 & - 17 \end{matrix}|$ and $|\begin{matrix} 1 & - 61 & - 23 \\ - 13 & - 26 & 26 \\ - 23 & 38 & - 17 \end{matrix}|$

$|\begin{matrix} 1 & - 61 & - 23 \\ - 13 & - 26 & 26 \\ - 23 & 38 & - 17 \end{matrix}|$ and $|\begin{matrix} 1 & - 61 & - 23 \\ - 13 & - 26 & 26 \\ - 23 & 38 & - 17 \end{matrix}|$

$|\begin{matrix} 1 & 61 & - 23 \\ 13 & - 26 & - 26 \\ - 23 & - 38 & - 17 \end{matrix}|$ and $|\begin{matrix} 1 & 61 & - 23 \\ 13 & - 26 & - 26 \\ - 23 & - 38 & - 17 \end{matrix}|$

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Detailed Solution

Here, $M_{11} = |\begin{matrix} 2 & - 5 \\ - 1 & 3 \end{matrix}| = 6 - 5 = 1$

[delete $1$ st row and $1$ st column]

$∴ C_{11} = {(- 1)}^{1 + 1} M_{11} = M_{11} = 1 M_{12} = |\begin{matrix} 7 & - 5 \\ 8 & 3 \end{matrix}| = 21 + 40 = 61$

[delete $1$ st row and $2$ nd column]

$∴ C_{12} = {(- 1)}^{1 + 2} M_{12} = - M_{12} = - 61 M_{13} = |\begin{matrix} 7 & 2 \\ 8 & - 1 \end{matrix}| = - 7 - 16 = - 23$

[delete $1$ st row and $3$ rd column]

$∴ C_{13} = {(- 1)}^{1 + 3} M_{13} = M_{13} = - 23 M_{21} = |\begin{matrix} 3 & 4 \\ - 1 & 3 \end{matrix}| = 9 + 4 = 13$

[delete $2$ nd row and $1$ st column]

$∴ C_{21} = {(- 1)}^{2 + 1} M_{21} = - M_{21} = - 13 M_{22} = |\begin{matrix} 2 & 4 \\ 8 & 3 \end{matrix}| = 6 - 32 = - 26$

[delete $2$ nd row and $2$ nd column]

$∴ C_{22} = {(- 1)}^{2 + 2} M_{22} = M_{22} = - 26 M_{23} = |\begin{matrix} 2 & 3 \\ 8 & - 1 \end{matrix}| = - 2 - 24 = - 26$

[delete $2$ nd row and $3$ rd column]

$∴ C_{23} = {(- 1)}^{2 + 3} M_{23} = - M_{23} = 26 M_{31} = |\begin{matrix} 3 & 4 \\ 2 & - 5 \end{matrix}| = - 15 - 8 = - 23$

[delete $3$ rd row and $1$ st column]

$∴ C_{31} = {(- 1)}^{3 + 1} M_{31} = M_{31} = - 23 M_{32} = |\begin{matrix} 2 & 4 \\ 7 & - 5 \end{matrix}| = - 10 - 28 = - 38$

[delete $3$ rd row and $2$ nd column]

$∴ C_{32} = {(- 1)}^{3 + 2} M_{32} = - M_{32} = 38 M_{33} = |\begin{matrix} 2 & 3 \\ 7 & 2 \end{matrix}| = 4 - 21 = - 17$

[delete $3$ rd row and $3$ rd column]

$∴ C_{33} = {(- 1)}^{3 + 3} M_{33} = M_{33} = - 17$

Hence, determinants of minors and cofactors are

$|\begin{matrix} 1 & 61 & - 23 \\ 13 & - 26 & - 26 \\ - 23 & - 38 & - 17 \end{matrix}|$ and $|\begin{matrix} 1 & - 61 & - 23 \\ - 13 & - 26 & 26 \\ - 23 & 38 & - 17 \end{matrix}|$ , respectively.