R S Aggarwal Solutions for Exercise 4: EXERCISE

Express the matrix $A$ as the sum of a symmetric and a skew-symmetric matrix, where $A=\left[\begin{array}{ccc}3& -1& 0\\ 2& 0& 3\\ 1& -1& 2\end{array}\right]$

Express the matrix $A=\left[\begin{array}{lll}3& 2& 5\\ 4& 1& 3\\ 0& 6& 7\end{array}\right]$ as sum of two matrices such that one is symmetric and the other is skew-symmetric.

For the following pair of matrices $A$ and $B$, verify that ${\left(AB\right)}^{\text{'}}=\left(B^{\text{'}}{A}^{\text{'}}\right)$:

$A=\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 4\\ 2& 5\end{array}\right]$

For the following pair of matrices $A$ and $B$, verify that ${\left(AB\right)}^{\text{'}}=\left(B^{\text{'}}{A}^{\text{'}}\right)$:

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

For the following pair of matrices $A$ and $B$, verify that ${\left(AB\right)}^{\text{'}}=\left(B^{\text{'}}{A}^{\text{'}}\right)$:

$A=\left[\begin{array}{c}-1\\ 2\\ 3\end{array}\right]$ and $B=\left[\begin{array}{ccc}-2& -1& -4\end{array}\right]$

Forthe following pair of matrices $A$ and $B$, verify that ${\left(AB\right)}^{\text{'}}=\left(B^{\text{'}}{A}^{\text{'}}\right)$:

$A=\left[\begin{array}{ccc}-1& 2& -3\\ 4& -5& 6\end{array}\right]$ and $B=\left[\begin{array}{cc}3& -4\\ 2& 1\\ -1& 0\end{array}\right]$

If $A=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]$, show that $A^{\text{'}}A=I$.

If matrix$A=\left[\begin{array}{ccc}1& 2& 3\end{array}\right]$, write $A{A}^{\text{'}}$.