Find the transpose of \(\begin{pmatrix}1&2\\3&4\end{pmatrix}\).

Find the transpose of \(\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}\).

If A is \(4\times 7\), state the size of \(A^T\).

For \(A=\begin{pmatrix}2&-1&5\\0&3&7\end{pmatrix}\), find the entry (\(A^T\))_32.

Find the transpose of the column vector \(\begin{pmatrix}3\\-1\\4\end{pmatrix}\).

Decide whether \(A=\begin{pmatrix}2&5\\5&-1\end{pmatrix}\) is symmetric.

Find all real x such that \(\begin{pmatrix}1&x\\3&4\end{pmatrix}\) is symmetric.

Verify (A + B)^T = \(A^T\) + \(B^T\) for \(A=\begin{pmatrix}1&2\\0&3\end{pmatrix}\) and \(B=\begin{pmatrix}4&-1\\5&2\end{pmatrix}\).

For \(A=\begin{pmatrix}1&2\\3&4\end{pmatrix}\) and \(B=\begin{pmatrix}0&5\\6&7\end{pmatrix}\), compute (\(AB\))^T and \(B^T\) \(A^T\) to verify the transpose product rule.

A student claims (\(AB\))^T = \(A^T\) \(B^T\) for all square matrices. Use \(A=\begin{pmatrix}1&1\\0&1\end{pmatrix}\) and \(B=\begin{pmatrix}1&0\\1&1\end{pmatrix}\) to diagnose the error.