If $A=\left[\begin{array}{cc}3& 5\\ -2& 0\\ 4& -6\end{array}\right]$, verify that $(2A{)}^{\text{'}}=2A^{\text{'}}$.

If $A=\left[\begin{array}{rrr}3& 2& -1\\ -5& 0& -6\end{array}\right]\text{ and }B=\left[\begin{array}{ccc}-4& -5& -2\\ 3& 1& 8\end{array}\right]$, verify that $(A+B{)}^{\text{'}}=(A^{\text{'}}+B^{\text{'}})$.

If $P=\left[\begin{array}{rr}3& 4\\ 2& -1\\ 0& 5\end{array}\right]$ and $Q=\left[\begin{array}{rr}7& -5\\ -4& 0\\ 2& 6\end{array}\right]$, verify that $(P+Q{)}^{\text{'}}=(P\text{'}+Q^{\text{'}}).$

If $A=\left[\begin{array}{ll}4& 1\\ 5& 8\end{array}\right]$ then show that $\left(A+A^{\text{'}}\right)$ is symmetric.

If $A=\left[\begin{array}{ll}3& -4\\ 1& -1\end{array}\right]$, show that $\left(A-A^{\text{'}}\right)$ is skew-symmetric.

Show that the matrix $A=\left[\begin{array}{ccc}0& a& b\\ -a& 0& c\\ -b& -c& 0\end{array}\right]$ is skew-symmetric.

Express the matrix $A=\left[\begin{array}{cc}2& 3\\ -1& 4\end{array}\right]$ as the sum of a symmetric matrix and a skew-symmetric matrix.

Express the matrix $A=\left[\begin{array}{ll}3& -4\\ 1& -1\end{array}\right]$ as the sum of a symmetric matrix and a skew-symmetric matrix.

Express the matrix $A=\left[\begin{array}{lll}-1& 5& 1\\ 2& 3& 4\\ 7& 0& 9\end{array}\right]$ as the sum of a symmetric and a skew-symmetric matrix.