On expanding by first row, the value of the determinant of 3 × 3 square matrix
Question:

On expanding by first row, the value of the determinant of 3 × 3 square matrix $A=\left[a_{i j}\right]$ is $a_{11} C_{11}+a_{12} C_{12}+a_{13} C_{13}$, where $\left[C_{i j}\right]$ is the cofactor of $a_{i j}$ in $A$. Write the expression for its value on expanding by second column.

Solution:

If $A=\left[\mathrm{a}_{\mathrm{i}}\right]$ is a square matrix of order $n$, then the sum of the products of elements of a row (or a column) with their cofactors is always equal to det (A). Therefore,

$\sum_{i=1}^{n} a_{i j} C_{i j}=|A|$ and $\sum_{j=1}^{n} a_{i j} C_{i j}=|A|$

Given: $|A|=\mathrm{a}_{11} \mathrm{C}_{11}+\mathrm{a}_{12} \mathrm{C}_{12}+\mathrm{a}_{13} \mathrm{C}_{13}$

[Expanding along $R_{1}$ ]

Now,

$|A|=a_{12} \mathrm{C}_{12}+\mathrm{a}_{22} \mathrm{C}_{22}+\mathrm{a}_{32} \mathrm{C}_{32}$    [Expanding along $\left.R_{2}\right] \quad\left[\mathrm{a}_{12}, \mathrm{a}_{22}\right.$ and $\mathrm{a}_{32}$ are elements of $\left.C_{2}\right]$