Question

if $A\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]$and $f(x)=(1+x)(1-x)$, then $f(A)$ is

• A. $-4\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$
• B. $-8\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$
• C. $4\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$
• D. None of these

Answer 1

Answer: (A)

Given, $f[x)=(1+x)(1-x)=1-x^2$
$\Rightarrow f(A)=I-A^2(\because$ Put $x=A$)
$\Rightarrow f\left(A\right)=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]-\left\{\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]\right\}$
$=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]-\left[\begin{array}{cc}5& 4\\ 4& 5\end{array}\right]=\left[\begin{array}{cc}-4& -4\\ -4& -4\end{array}\right]$
$=-4\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$