Question

Which of the following are the properties of transpose :-

• A. ${\left(A^T\right)}^T=A$
• B. ${(A\pm B)}^T=A^T\pm B^T$
• C. ${\left(A^T\right)}^2={\left(A^2\right)}^T$
• D. ${\left(\mathrm{AB}\right)}^T=B^T{A}^T$

Answer 1

Answer: (A), (B), (C), (D)

${\left(\mathrm{AB}\right)}^T=B^T{A}^T$

$\mathrm{Let}A=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\mathrm{and}B=\left(\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]\left(1\right){\left(A^T\right)}^T={\left(\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]}^T=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]=A\left(2\right){(A\pm B)}^T=A^T\pm B^T{(A+B)}^T={\left(\begin{array}{cc}3& 5\\ 7& 9\end{array}\right]}^T=\left(\begin{array}{cc}3& 7\\ 5& 9\end{array}\right]A^T=\left(\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]B^T=\left(\begin{array}{cc}2& 4\\ 3& 5\end{array}\right]A^T+B^T=\left(\begin{array}{cc}3& 7\\ 5& 9\end{array}\right]\Rightarrow {(A\pm B)}^T=A^T\pm B^T\left(3\right){\left(A^2\right)}^T={\left({\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]}^2\right)}^T={\left(\begin{array}{cc}7& 10\\ 15& 22\end{array}\right]}^T=\left(\begin{array}{cc}7& 15\\ 10& 22\end{array}\right]{\left(A^T\right)}^2={\left(\right(\begin{array}{cc}1& 3\\ 2& 4\end{array}\left]\right)}^2=\left(\begin{array}{cc}7& 15\\ 10& 22\end{array}\right]={\left(A^2\right)}^T\left(4\right){\left(\mathrm{AB}\right)}^T={\left(\begin{array}{cc}10& 13\\ 22& 29\end{array}\right]}^T=\left(\begin{array}{cc}10& 22\\ 13& 29\end{array}\right]B^T{A}^T=\left(\begin{array}{cc}2& 4\\ 3& 5\end{array}\right]\left(\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]=\left(\begin{array}{cc}10& 22\\ 13& 29\end{array}\right]={\left(\mathrm{AB}\right)}^T$