Question

If $A = \begin{pmatrix}2&-1\\ -7&4\end{pmatrix}$ and $B = \begin{pmatrix}4&1\\ 7&2\end{pmatrix}$ then which statement is true ?

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

Answer 1

Answer: (D)

Here $AA^{T} = \begin{pmatrix}2&-1\\ -7&4\end{pmatrix}\begin{pmatrix}2&-7\\ -1&4\end{pmatrix}\ne \begin{pmatrix}1&0\\ 0&1\end{pmatrix}$
$\left(BB^{T}\right)_{11}= \left(4\right)^{2} + \left(1\right)^{2}\ne1$
$\left(AB\right)_{11} = 8-7 = 1, \left(BA\right)_{11} = 8-7=1$
Now, $AB=\begin{pmatrix}2&-1\\ -7&4\end{pmatrix} \begin{pmatrix}4&1\\ 7&2\end{pmatrix}$
$= \begin{pmatrix}8-7&2-2\\ -28+28&-7+8\end{pmatrix} = \begin{pmatrix}1&0\\ 0&1\end{pmatrix}$
$\therefore \left(AB\right)^{T} = \begin{pmatrix}1&0\\ 0&1\end{pmatrix}$