If $A=\begin{bmatrix}2 & -3\\3 & -2\\-1 & 4\end{bmatrix}$, $B=\begin{bmatrix}-3 & 4 & 1\\2 & -1 & -3\end{bmatrix}$, Verify $(A+B^T)^T=A^T+2B$.

If A= 2 -3; 3 -2; -1 4 , B= -3 4 1; 2 -1 -3 , Verify (A+BT)T=AT+2B.

Miscellaneous Exercise 4(B) · Q187

A=\begin{bmatrix}2 & -3\\3 & -2\\-1 & 4\end{bmatrix}

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$A=\begin{bmatrix}2 & -3\\3 & -2\\-1 & 4\end{bmatrix}$ (3x2), $B=\begin{bmatrix}-3 & 4 & 1\\2 & -1 & -3\end{bmatrix}$ (2x3), so $B^T=\begin{bmatrix}-3 & 2\\4 & -1\\1 & -3\end{bmatrix}$ (3x2).

$A+B^T=\begin{bmatrix}-1 & -1\\7 & -3\\0 & 1\end{bmatrix}$ , so $(A+B^T)^T=\begin{bmatrix}-1 & 7 & 0\\-1 & -3 & 1\end{bmatrix}$ .

$A^T=\begin{bmatrix}2 & 3 & -1\\-3 & -2 & 4\end{bmatrix}$ . Compare to the printed RHS $A^T+2B$ : $2B=\begin{bmatrix}-6 & 8 & 2\\4 & -2 & -6\end{bmatrix}$ , so $A^T+2B=\begin{bmatrix}-4 & 11 & 1\\1 & -4 & -2\end{bmatrix}$ — this does NOT match $(A+B^T)^T$ . …

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Q.If $A=\begin{bmatrix}2 & -3\\3 & -2\\-1 & 4\end{bmatrix}$ , $B=\begin{bmatrix}-3 & 4 & 1\\2 & -1 & -3\end{bmatrix}$ , Verify $(A+B^T)^T=A^T+2B$ .

Concept understanding — Transpose of a Matrix

Q.If A=[2−33−2−14]A=\begin{bmatrix}2 & -3\\3 & -2\\-1 & 4\end{bmatrix}A=​23−1​−3−24​​, B=[−3412−1−3]B=\begin{bmatrix}-3 & 4 & 1\\2 & -1 & -3\end{bmatrix}B=[−32​4−1​1−3​], Verify (A+BT)T=AT+2B(A+B^T)^T=A^T+2B(A+BT)T=AT+2B.

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Q.If $A=\begin{bmatrix}2 & -3\\3 & -2\\-1 & 4\end{bmatrix}$ , $B=\begin{bmatrix}-3 & 4 & 1\\2 & -1 & -3\end{bmatrix}$ , Verify $(A+B^T)^T=A^T+2B$ .

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