#### Explain solution for RD Sharma Class 12 Chapter 5 Determinants Exercise Very short Answers Question 17 maths textbook solution.

Answer : $a_{12} c_{12}+a_{22} c_{22}+a_{32} c_{32}$

Hint : Here we use basic concept of determinant of matrix and cofactor

Given: $A=\left [ a_{ij} \right ]$ is $a_{11}c_{11}+a_{12}c_{12}+a_{13}c_{13}$

where $\left [ c_{ij} \right ]$ is the cofactor

Solution :

If $A=\left[a_{i j}\right]$ is a square matrix of order n then the sum of the products of elements of a row with their cofactor is always equal to det(A).

Therefore,

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

$|A|=a_{11} c_{11}+a_{12} c_{12}+a_{13} c_{13} \quad\left[\text { expanding along } \mathrm{R}_{1}\right]$

$|A|=a_{12} c_{12}+a_{22} c_{22}+a_{32} c_{32}\left[\begin{array}{l} {\left[\text { expanding along } \mathrm{R}_{2}\right]} \\ {\left[a_{12}, a_{22} \text { and } a_{32} \text { are the elements of } \mathrm{C}_{2}\right.} \end{array}\right.$