Question

If $A=\left(\begin{array}{cc}2& -1\\ -7& 4\end{array}\right)$ and $B=\left(\begin{array}{cc}4& 1\\ 7& 2\end{array}\right)$ then which statement is true ?

• A. $A{A}^T=I$
• B. $B{B}^T=I$
• C. $AB\ne BA$
• D. ${\left(AB\right)}^T=I$

Answer 1

Answer: (D)

Here $A{A}^T=\left(\begin{array}{cc}2& -1\\ -7& 4\end{array}\right)\left(\begin{array}{cc}2& -7\\ -1& 4\end{array}\right)\ne \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$
${\left(B{B}^T\right)}_{11}={\left(4\right)}^2+{\left(1\right)}^2\ne 1$
${\left(AB\right)}_{11}=8-7=1,{\left(BA\right)}_{11}=8-7=1$
Now, $AB=\left(\begin{array}{cc}2& -1\\ -7& 4\end{array}\right)\left(\begin{array}{cc}4& 1\\ 7& 2\end{array}\right)$
$=\left(\begin{array}{cc}8-7& 2-2\\ -28+28& -7+8\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$
$\therefore {\left(AB\right)}^T=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$