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Mathematics Question on Matrices

if A[12 21]A \begin{bmatrix}1&2\\\ 2&1\end{bmatrix}and f(x)=(1+x)(1−x)f(x) = (1+x) (1 - x), then f(A)f(A) is

A

−4[11 11]-4 \begin{bmatrix}1&1\\\ 1&1\end{bmatrix}

B

−8[11 11]-8 \begin{bmatrix}1&1\\\ 1&1\end{bmatrix}

C

4[11 11]4 \begin{bmatrix}1&1\\\ 1&1\end{bmatrix}

D

None of these

Answer

−4[11 11]-4 \begin{bmatrix}1&1\\\ 1&1\end{bmatrix}

Explanation

Solution

Given, f[x)=(1+x)(1−x)=1−x2f[x) = (1 + x) (1 - x) = 1 - x^2 ⇒f(A)=I−A2(∵\Rightarrow f(A) = I - A^2 (\because Put x=Ax = A) \Rightarrow f\left(A\right) = \begin{bmatrix}1&0\\\ 0&1\end{bmatrix}-\left\\{\begin{bmatrix}1&2\\\ 2&1\end{bmatrix}\begin{bmatrix}1&2\\\ 2&1\end{bmatrix}\right\\} =[10 01]−[54 45]=[−4−4 −4−4]=\begin{bmatrix}1&0\\\ 0&1\end{bmatrix}-\begin{bmatrix}5&4\\\ 4&5\end{bmatrix}=\begin{bmatrix}-4&-4\\\ -4&-4\end{bmatrix} =−4[11 11] = -4 \begin{bmatrix}1&1\\\ 1&1\end{bmatrix}