#### Please Solve RD Sharma Class 12 Chapter 4 Algebra of Matrices Exercise Very Short Asnwer Question 12 Maths Textbook Solution.

Answer: $\lambda=8$

Hint: Here, we use the basic concept of Algebra

Given:

$A=\begin{bmatrix} 1 & 1\\ 1 & 1 \end{bmatrix}$

$A^{^{4}}=\lambda A$

Solution:

$A^{^{4}}= A^{2}\times A^{2}$

$A^{2}=\begin{bmatrix} 1 & 1\\ 1 & 1 \end{bmatrix}\times \begin{bmatrix} 1 & 1\\ 1 & 1 \end{bmatrix}$

$A^{2}=\begin{bmatrix} 1 +1& 1+1\\ 1+1 & 1+1 \end{bmatrix}=\begin{bmatrix} 2 & 2\\ 2 & 2 \end{bmatrix}$

$A^{2}\times A^{2}=\begin{bmatrix} 2 & 2\\ 2 & 2 \end{bmatrix}\begin{bmatrix} 2 & 2\\ 2 & 2 \end{bmatrix}$

$=\begin{bmatrix} 4+4 & 4+4\\ 4+4 & 4+4 \end{bmatrix}=\begin{bmatrix} 8 & 8\\ 8 & 8 \end{bmatrix}$

$A^{4}=\begin{bmatrix} 8 & 8\\ 8 & 8 \end{bmatrix}$

$A^{^{4}}=\lambda A$

$\begin{bmatrix} 8 & 8\\ 8 & 8 \end{bmatrix}=\lambda \begin{bmatrix} 1 & 1\\ 1 & 1 \end{bmatrix}$

$\begin{bmatrix} 8 & 8\\ 8 & 8 \end{bmatrix}= \begin{bmatrix} \lambda & \lambda\\ \lambda & \lambda \end{bmatrix}$

So,$\lambda=8$