# Get Answers to all your Questions

#### Explain solution for RD Sharma Class 12 Chapter 5 Determinants Exercise Very short Answers Question 19 maths textbook solution.

Hint: Here we use basic concept of determinant of matrix

Given: \begin{aligned} &A \text { is } 3 \times 3 \text { matrix }\\ &\mathrm{S} 0, \mathrm{n}=3,|A|=5 \text { and } \mathrm{C}_{i j} \text { is cofactor of } a_{i} \end{aligned}

Solution :

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

and cofactor $\mathrm{C}_{i j}=(-1)^{i+j}\left|m_{i j}\right|$

So, here

$a_{11} C_{21}+a_{12} C_{12}+a_{13} C_{23}$

So

\begin{aligned} &C_{21}=(-1)^{2+1}\left|a_{12} a_{33}-a_{13} a_{32}\right| \\ &C_{22}=(-1)^{2+2}\left|a_{11} a_{33}-a_{13} a_{31}\right| \\ &C_{23}=(-1)^{2+3}\left|a_{11} a_{32}-a_{12} a_{31}\right| \end{aligned}

\begin{aligned} &\rightarrow a_{11} A_{21}+a_{12} A_{22}+a_{13} A_{22} \\ &\rightarrow-a_{11} a_{12} a_{33}+a_{11} a_{13} a_{32}+a_{11} a_{12} a_{33}-a_{12} a_{13} a_{31}-a_{11} a_{13} a_{32}+a_{12} a_{13} a_{31} \\ &=0 \end{aligned}