Minors And Cofactors - Practice Questions & MCQ

The cofactor of the matrix or determinant is the same as the minor of the matrix or determinant but the only difference is of sign, if $\mathrm{i}+\mathrm{j}$ is even then the cofactor $=$ minor if $\mathrm{i}+\mathrm{j}$ is odd then cofactor $=-\mathrm{minor}$,

$
\begin{aligned}
\mathrm{C}_{\mathrm{ij}} & =(-1)^{i+j} \mathrm{M}_{i j} \\
& =\left\{\begin{array}{cc}
M_{i j} & \text { if } \mathrm{i}+\mathrm{j} \text { is an even integer } \\
-M_{i j} & \text { if } \mathrm{i}+\mathrm{j} \text { is an odd integer }
\end{array}\right.
\end{aligned}
$
Or we can write, $\mathrm{C}_{\mathrm{ij}}$ is co - factor of $\mathrm{a}_{\mathrm{ij}}$
For example,

$
\Delta=\left|\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right|
$

then, minor of the element $\mathrm{a}_{21}$ is

$
M_{21}=\left|\begin{array}{ll}
a_{12} & a_{13} \\
a_{32} & a_{33}
\end{array}\right| \text { and that of } a_{32} \text { is } M_{32}=\left|\begin{array}{ll}
a_{11} & a_{13} \\
a_{21} & a_{23}
\end{array}\right| .
$
Cofactor of the element $a_{21}$ is

$
C_{21}=(-1)^{2+1} M_{21}=-\left|\begin{array}{ll}
a_{12} & a_{13} \\
a_{32} & a_{33}
\end{array}\right|
$

If we expand the determinant $\Delta$ along the first row, then value of $\Delta$ in terms of minors is $\mathbf{a}_{11} \mathbf{M}_{11}-\mathbf{a}_{12} \mathrm{M}_{12}-\mathbf{a}_{13} \mathbf{M}_{13}$.