On expanding by first row, the value of the determinant of 3 × 3 square matrix
If
$\sum_{i=1}^{n} a_{i j} C_{i j}=|A|$ 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 $\left.C_{2}\right]$
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