# then compute and. Also, verify that

Question:

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

then compute $(A+B)$ and $(B-C)$. Also, verify that $A+(B-C)=(A+B)-C$

Solution:

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

$=\left[\begin{array}{ccc}1+3 & 2-1 & -3+2 \\ 5+4 & 0+2 & 2+5 \\ 1+2 & -1+0 & 1+3\end{array}\right]=\left[\begin{array}{ccr}4 & 1 & -1 \\ 9 & 2 & 7 \\ 3 & -1 & 4\end{array}\right]$

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

$=\left[\begin{array}{ccc}3-4 & -1-1 & 2-2 \\ 4-0 & 2-3 & 5-2 \\ 2-1 & 0-(-2) & 3-3\end{array}\right]=\left[\begin{array}{rrr}-1 & -2 & 0 \\ 4 & -1 & 3 \\ 1 & 2 & 0\end{array}\right]$

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

$=\left[\begin{array}{ccc}1+(-1) & 2+(-2) & -3+0 \\ 5+4 & 0+(-1) & 2+3 \\ 1+1 & -1+2 & 1+0\end{array}\right]=\left[\begin{array}{ccc}0 & 0 & -3 \\ 9 & -1 & 5 \\ 2 & 1 & 1\end{array}\right]$

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

$=\left[\begin{array}{ccc}4-4 & 1-1 & -1-2 \\ 9-0 & 2-3 & 7-2 \\ 3-1 & -1-(-2) & 4-3\end{array}\right]=\left[\begin{array}{ccc}0 & 0 & -3 \\ 9 & -1 & 5 \\ 2 & 1 & 1\end{array}\right]$

$=\left[\begin{array}{ccc}4-4 & 1-1 & -1-2 \\ 9-0 & 2-3 & 7-2 \\ 3-1 & -1-(-2) & 4-3\end{array}\right]=\left[\begin{array}{ccc}0 & 0 & -3 \\ 9 & -1 & 5 \\ 2 & 1 & 1\end{array}\right]$

Hence, we have verified that $A+(B-C)=(A+B)-C$.