# If and, then verify that

Question:

If $A=\left[\begin{array}{rrr}-1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1\end{array}\right]$ and $B=\left[\begin{array}{rrr}-4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1\end{array}\right]$, then verify that

(i) $(A+B)^{\prime}=A^{\prime}+B^{\prime}$

(ii) $(A-B)^{\prime}=A^{\prime}-B^{\prime}$

Solution:

We have:

$A^{\prime}=\left[\begin{array}{rrr}-1 & 5 & -2 \\ 2 & 7 & 1 \\ 3 & 9 & 1\end{array}\right], B^{\prime}=\left[\begin{array}{rrr}-4 & 1 & 1 \\ 1 & 2 & 3 \\ -5 & 0 & 1\end{array}\right]$

(i)

$A+B=\left[\begin{array}{rrr}-1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1\end{array}\right]+\left[\begin{array}{rrr}-4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1\end{array}\right]=\left[\begin{array}{rrr}-5 & 3 & -2 \\ 6 & 9 & 9 \\ -1 & 4 & 2\end{array}\right]$

$\therefore(A+B)^{\prime}=\left[\begin{array}{rrr}-5 & 6 & -1 \\ 3 & 9 & 4 \\ -2 & 9 & 2\end{array}\right]$

$A^{\prime}+B^{\prime}=\left[\begin{array}{rrr}-1 & 5 & -2 \\ 2 & 7 & 1 \\ 3 & 9 & 1\end{array}\right]+\left[\begin{array}{rrr}-4 & 1 & 1 \\ 1 & 2 & 3 \\ -5 & 0 & 1\end{array}\right]=\left[\begin{array}{rrr}-5 & 6 & -1 \\ 3 & 9 & 4 \\ -2 & 9 & 2\end{array}\right]$

Hence, we have verified that $(A+B)^{\prime}=A^{\prime}+B^{\prime}$

(ii)

$A-B=\left[\begin{array}{rrr}-1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1\end{array}\right]-\left[\begin{array}{rrr}-4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1\end{array}\right]=\left[\begin{array}{rrr}3 & 1 & 8 \\ 4 & 5 & 9 \\ -3 & -2 & 0\end{array}\right]$

$\therefore(A-B)^{\prime}=\left[\begin{array}{rrr}3 & 4 & -3 \\ 1 & 5 & -2 \\ 8 & 9 & 0\end{array}\right]$

$A^{\prime}-B^{\prime}=\left[\begin{array}{rrr}-1 & 5 & -2 \\ 2 & 7 & 1 \\ 3 & 9 & 1\end{array}\right]-\left[\begin{array}{rrr}-4 & 1 & 1 \\ 1 & 2 & 3 \\ -5 & 0 & 1\end{array}\right]=\left[\begin{array}{rrr}3 & 4 & -3 \\ 1 & 5 & -2 \\ 8 & 9 & 0\end{array}\right]$

Hence, we have verified that $(A-B)^{\prime}=A^{\prime}-B^{\prime}$.