Provide Solution for RD Sharma Class 12 Chapter 6 Adjoint and Inverse Matrix Exercise 6.1 Question 38

\begin{aligned} &A \times \operatorname{Adj}(A)=27\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \\ & \end{aligned}

Hint:

Here, we use basic concept of determinant and inverse of matrix

$A^{-1}=\frac{1}{|A|} \times A d j(A)$

Given:

$A=\left|\begin{array}{ccc} -1 & -2 & -2 \\ 2 & 1 & -2 \\ 2 & -2 & 1 \end{array}\right|$

Solution:

\begin{aligned} &|A|=\left|\begin{array}{ccc} -1 & -2 & -2 \\ 2 & 1 & -2 \\ 2 & -2 & 1 \end{array}\right| \\ &|A|=-1(1-4)+2(2+4)-2(-4-2) \\ &=3+12+12 \\ &|A|=27 \end{aligned}

Cofactor of A

\begin{aligned} &C_{11}=-3, C_{21}=6, C_{31}=6 \\ &C_{12}=-6, C_{22}=3, C_{32}=-6 \\ &C_{13}=-6, C_{23}=-6, C_{33}=3 \end{aligned}

$\operatorname{Adj}(A)=\left[\begin{array}{ccc} -3 & 6 & 6 \\ -6 & 3 & -6 \\ 6 & 6 & 3 \end{array}\right]$

\begin{aligned} & \\ &A \times \operatorname{Adj}(A)=\left[\begin{array}{ccc} -1 & -2 & -2 \\ 2 & 1 & -2 \\ 2 & -2 & 1 \end{array}\right]\left[\begin{array}{ccc} -3 & 6 & 6 \\ -6 & 3 & -6 \\ 6 & 6 & 3 \end{array}\right]=\left[\begin{array}{ccc} 27 & 0 & 0 \\ 0 & 27 & 0 \\ 0 & 0 & 27 \end{array}\right] \end{aligned}

\begin{aligned} &A \times \operatorname{Adj}(A)=27\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \\ & \end{aligned}

$A \times A d j(A)=|A| I$