#### Please Solve RD Sharma Class 12 Chapter 6 Adjoint and Inverese of Matrices Exercise 6.1 Question 3 Maths Textbook Solution.

Proved  $A(\operatorname{Adj}(A))=0$

Hint:

Here, we have to use advance method of finding adjoint of matrix.

Given:

$A=\left[\begin{array}{ccc} 1 & -1 & 1 \\ 2 & 3 & 0 \\ 18 & 2 & 10 \end{array}\right]$

Solution:

We know that,

$A \times \operatorname{Adj}(A)=|A| I$  Then,

$\left | A \right |I= 0$                       (1)

From the equation (1)

$\left | A \right |I= 0$

$\left|\begin{array}{ccc} 1 & -1 & 1 \\ 2 & 3 & 0 \\ 18 & 2 & 10 \end{array}\right| \times 1=0$                                                      since, $|I| =1$

$1\left|\begin{array}{rr} 3 & 0 \\ 2 & 10 \end{array}\right|-(-1)\left|\begin{array}{cc} 2 & 0 \\ 18 & 10 \end{array}\right|+1\left|\begin{array}{cc} 2 & 3 \\ 18 & 2 \end{array}\right|=0$

\begin{aligned} &0=1(30)+(20-0)+(4-54) \\ &0=30+20+4-54 \\ &0=54-54 \\ &0=0 \end{aligned}

So, $A \times \operatorname{Adj}(A)=|A| I=0$