Q.Show with usual notations that for any matrix $A = [a_{ij}]_{3\times3}$: $a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13} = |A|$

A minor is defined for each element of the determinant, and a $3\times3$ determinant has 9 elements.

…

The cofactor of the entry -3 (at row 3, column 2) is computed as C32=(-1)^5 times its minor, giving 2 - not any of the listed options.

The entry $-3$ sits at position $a_{32}$ (row 3, column 2) in $\begin{vmatrix}0&1&-2\\-1&0&2\\2&-3&0\end{vmatrix}$.

Minor $M_{32}$ = determinant left after deleting row 3 and column 2:

$M_{32}=\begin{vmatrix}0&-2\\-1&2\end{vmatrix} = (0)(2)-(-2)(-1) = 0-2=-2$

…

Expansion of a determinant along a column, using cofactors of that column's entries.

The value of a determinant can be obtained by expanding along any row or column, using the cofactors of that row/column's entries. Expanding $\Delta$ along the first column ($a_{11},a_{21},a_{31}$) using their respective cofactors $A_{11},A_{21},A_{31}$:

$\Delta = a_{11}A_{11}+a_{21}A_{21}+a_{31}A_{31}$ …

Standard definition from the Determinants chapter.

Minor of an element: For a determinant, the minor $M_{ij}$ of the element $a_{ij}$ is the determinant obtained after deleting the $i$-th row and the $j$-th column, i.e. the row and column in which $a_{ij}$ lies. It is a determinant of order one less than the order of the original determinant.

…

Cofactor $C_{ij}=(-1)^{i+j}M_{ij}$; delete row 3 and column 1, then evaluate the $2\times2$ minor.

$\begin{vmatrix}1 & 3 & 5\\ -1 & 2 & 3\\ 2 & 6 & -7\end{vmatrix}$

To find $C_{31}$, delete row 3 and column 1: …

Element 8 is at row 1, column 1; its cofactor is $(-1)^{1+1}$ times its minor, giving $+14$.

For the determinant $\begin{vmatrix} 8 & 10 \\ 12 & 14 \end{vmatrix}$, the element $8$ lies in the first row and first column, so $i=1,\ j=1$.

…

The minor of an element is the determinant left after deleting its row and column.

In $\begin{vmatrix}1 & 3 & 2\\ 8 & 6 & 3\\ 9 & 5 & 4\end{vmatrix}$, the element 6 is in row 2, column 2.

…

State the definition of a determinant.

To every square matrix $A=[a_{ij}]$ of order $n$, we can associate a number (real or complex), called the determinant of the square matrix $A$, denoted by $\det(A)$ or $|A|$.

For example, for a $2\times2$ matrix $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$,

$|A|=ad-bc$

…

The minor of an element is the determinant left after deleting that element's row and column; for a $2\times2$ matrix each minor is just the single opposite-corner entry.

Given $\begin{vmatrix}2 & -4\\ 0 & 3\end{vmatrix}$, the minor $M_{ij}$ is obtained by deleting row $i$ and column $j$ and taking the determinant of what remains (here, a single number).

$M_{11}$ (delete row 1, col 1): remaining entry is $3$, so $M_{11}=3$

…