Question

If $A=\left[\begin{array}{cc}5& 2\\ 2& 1\end{array}\right]$ and matrix $B$ is such that $A{B}^{T}A=B$ and $B^{T}AB=I$ where $I$ is unit matrix of order $2$, then matrix $B^2$ is

• A. $\left[\begin{array}{cc}29& 12\\ 12& 5\end{array}\right]$
• B. $\left[\begin{array}{cc}1& -2\\ -2& 5\end{array}\right]$
• C. $\left[\begin{array}{cc}-1& 2\\ 2& -5\end{array}\right]$
• D. $\left[\begin{array}{cc}5& 2\\ 2& 1\end{array}\right]$

Answer 1

The correct option is D $\left[\begin{array}{cc}5& 2\\ 2& 1\end{array}\right]$

$A{B}^{T}A=B$
$\Rightarrow A^{-1}A{B}^{T}A=A^{-1}B$
$\Rightarrow B^{T}A=A^{-1}B\Rightarrow B^{T}AB=A^{-1}BB$
$\Rightarrow I=A^{-1}{B}^2\{\because B^{T}AB=I\}$
$\Rightarrow B^2=A$