Question

Consider two matrices A and B. $(A+B{)}^T$ = __________

• A. (AB)^T
• B. AB
• C. A^T + B^T
• D. BA

Answer 1

The correct option is C

A^T + B^T

$(A+B{)}^T$ = $(A{)}^T$ +​ $(B{)}^T$

The matrix obtained by interchanging the rows and columns of a matrix is said to be the transpose of that matrix.

Consider an example

$A=\left[\begin{array}{cc}1& 2\\ 2& 3\end{array}\right]andB=\left[\begin{array}{cc}4& 1\\ 2& 0\end{array}\right]$

$A+B=\left[\begin{array}{cc}5& 3\\ 4& 3\end{array}\right]$

$(A+B{)}^T=\left[\begin{array}{cc}5& 4\\ 3& 3\end{array}\right]$

$A^T=\left[\begin{array}{cc}1& 2\\ 2& 3\end{array}\right]and{B}^T=\left[\begin{array}{cc}4& 2\\ 1& 0\end{array}\right]$

$A^T+B^T=\left[\begin{array}{cc}5& 4\\ 3& 3\end{array}\right]$

Therefore, $(A+B{)}^T=A^T+B^T$.