Question

If $A=\left[\begin{array}{cc}3& 2\\ 1& 4\end{array}\right]$ and $B=\left[\begin{array}{cc}-1& 2\\ -1& 1\end{array}\right]$, then correct statement is

• A. $AB=BA$
• B. $A{A}^T=A^2$
• C. $AB=B^2$
• D. None of these

Answer 1

Answer: (D)

None of these

For $A=\left[\begin{array}{cc}3& 2\\ 1& 4\end{array}\right]$ and $B=\left[\begin{array}{cc}-1& 2\\ -1& 1\end{array}\right]$
$AB=\left[\begin{array}{cc}3& 2\\ 1& 4\end{array}\right]\left[\begin{array}{cc}-1& 2\\ -1& 1\end{array}\right]=\left[\begin{array}{cc}-5& 8\\ -5& 6\end{array}\right]$
$BA=\left[\begin{array}{cc}-1& 2\\ -1& 1\end{array}\right]\left[\begin{array}{cc}3& 2\\ 1& 4\end{array}\right]=\left[\begin{array}{cc}-1& 6\\ -2& 2\end{array}\right]$
$A^T=\left[\begin{array}{cc}3& 1\\ 2& 4\end{array}\right]$
$A{A}^T=\left[\begin{array}{cc}3& 2\\ 1& 4\end{array}\right]\left[\begin{array}{cc}3& 1\\ 2& 4\end{array}\right]=\left[\begin{array}{cc}13& 11\\ 11& 17\end{array}\right]$
$A^2=\left[\begin{array}{cc}3& 2\\ 1& 4\end{array}\right]\left[\begin{array}{cc}3& 2\\ 1& 4\end{array}\right]=\left[\begin{array}{cc}11& 14\\ 7& 18\end{array}\right]$
$B^2=\left[\begin{array}{cc}-1& 2\\ -1& 1\end{array}\right]\left[\begin{array}{cc}-1& 2\\ -1& 1\end{array}\right]=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$

As all the above options are not holding true so the correct option is D