Question

If $A^{\prime }=\left[\begin{array}{c}3\\ -1\\ 0\end{array}\begin{array}{c}4\\ 2\\ 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}-1& 2& 1\\ 1& 2& 3\end{array}\right]$, then verify that:

$\left(i\right){(A+B)}^{\prime }=A^{\prime }+B^{\prime }$
$\left(ii\right){(A-B)}^{\prime }=A^{\prime }-B^{\prime }$

Answer 1

(i) It is known that $A={\left(A^{\prime }\right)}^{\prime }$
Therefore, we have:
$A=\left[\begin{array}{ccc}3& -1& 0\\ 4& 2& 1\end{array}\right]$
$B^{\prime }=\left[\begin{array}{c}-1\\ 2\\ 1\end{array}\begin{array}{c}1\\ 2\\ 3\end{array}\right]$
$A+B=\left[\begin{array}{ccc}3& -1& 0\\ 4& 2& 1\end{array}\right]+\left[\begin{array}{ccc}-1& 2& 1\\ 1& 2& 3\end{array}\right]$
$\therefore {(A+B)}^{\prime }=\left[\begin{array}{c}2\\ 1\\ 1\end{array}\begin{array}{c}5\\ 4\\ 4\end{array}\right]$
$A^{\prime }+B^{\prime }=\left[\begin{array}{c}3\\ -1\\ 0\end{array}\begin{array}{c}4\\ 2\\ 1\end{array}\right]+\left[\begin{array}{c}-1\\ 2\\ 1\end{array}\begin{array}{c}1\\ 2\\ 3\end{array}\right]=\left[\begin{array}{c}2\\ 1\\ 1\end{array}\begin{array}{c}5\\ 4\\ 4\end{array}\right]$
Thus, we have verified that ${(A+B)}^{\prime }=A^{\prime }+B^{\prime }$.
(ii) $A-B=\left[\begin{array}{ccc}3& -1& 0\\ 4& 2& 1\end{array}\right]-\left[\begin{array}{ccc}-1& 2& 1\\ 1& 2& 3\end{array}\right]=\left[\begin{array}{ccc}4& -3& -1\\ 3& 0& -2\end{array}\right]$
$\therefore {(A-B)}^{\prime }=\left[\begin{array}{c}4\\ -3\\ -1\end{array}\begin{array}{c}3\\ 0\\ -2\end{array}\right]$
$A^{\prime }-B^{\prime }=\left[\begin{array}{c}3\\ -1\\ 0\end{array}\begin{array}{c}4\\ 2\\ 1\end{array}\right]-\left[\begin{array}{c}-1\\ 2\\ 1\end{array}\begin{array}{c}1\\ 2\\ 3\end{array}\right]=\left[\begin{array}{c}4\\ -3\\ -1\end{array}\begin{array}{c}3\\ 0\\ -2\end{array}\right]$
Thus, we have verified that ${(A-B)}^{\prime }=A^{\prime }-B^{\prime }$