#### Need solution for RD Sharma maths class 12 chapter Algebra of matrices exercise 4.3 question 37math

Answer:  Hence proved$f(A)=0$

Hint: I is an identity matrix.

Matrix multiplication is only possible, when the number of columns of first matrix is equal to the number of rows of second matrix.

Given: $A=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]$and $f(x)=x^{2}-2 x-3$

To show that $f(A)=0$

Substitute x=A in f(x) we get

$f(A)=A^{2}-2 A-3 I---i\\\\ f(A)=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]-2\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]-3\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\\\\ =\left[\begin{array}{ll}1+4 & 2+2 \\ 2+2 & 4+1\end{array}\right]-\left[\begin{array}{ll}2 & 4 \\ 4 & 2\end{array}\right]-\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right] =\left[\begin{array}{ll}5 & 4 \\ 4 & 5\end{array}\right]-\left[\begin{array}{ll}2 & 4 \\ 4 & 2\end{array}\right]-\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right]\\\\ =\left[\begin{array}{lll}5-2-3 & 4-4-0 \\ 4-4-0 & 5-2-3\end{array}\right]\\\\ =\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]\\\\ =0\\\\$

So, $f (A) = 0$

Hence, proved