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If $A=\left[\begin{array}{ccc}1 & 2 & -3 \\ 5 & 0 & 2 \\ 1 & -1 & 1\end{array}\right], B=\left[\begin{array}{ccc}3 & -1 & 2 \\ 4 & 2 & 5 \\ 2 & 0 & 3\end{array}\right]$
$$
C=\left[\begin{array}{ccc}
4 & 1 & 2 \\
0 & 3 & 2 \\
1 & -2 & 3
\end{array}\right]
$$
then compute $(A+B)$ and $(B-C)$. Also verify that $A+(B-C)=(A+B)-C$
$$
C=\left[\begin{array}{ccc}
4 & 1 & 2 \\
0 & 3 & 2 \\
1 & -2 & 3
\end{array}\right]
$$
then compute $(A+B)$ and $(B-C)$. Also verify that $A+(B-C)=(A+B)-C$
Solution:
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Verified Answer
$\mathrm{A}+\mathrm{B}=\left[\begin{array}{ccc}4 & 1 & -1 \\ 9 & 2 & 7 \\ 3 & -1 & 4\end{array}\right]$
$$
\mathrm{B}-\mathrm{C}=\left[\begin{array}{ccc}
-1 & -2 & 0 \\
4 & -1 & 3 \\
1 & 2 & 0
\end{array}\right]
$$
L.H.S. A $+(\mathrm{B}-\mathrm{C})$ putting the value of $(\mathrm{B}-\mathrm{C})$ from (ii)
$$
\left[\begin{array}{ccc}
1 & 2 & -3 \\
5 & 0 & 2 \\
1 & -1 & 1
\end{array}\right]+\left[\begin{array}{ccc}
-1 & -2 & 0 \\
4 & -1 & 3 \\
1 & 2 & 0
\end{array}\right]=\left[\begin{array}{ccc}
0 & 0 & -3 \\
9 & -1 & 5 \\
2 & 1 & 1
\end{array}\right] \ldots
$$
R.H.S : $(A+B)-C$ putting the value of $(A+B)$ from (i)
$$
\left[\begin{array}{ccc}
4 & 1 & -1 \\
9 & 2 & 7 \\
3 & -1 & 4
\end{array}\right]-\left[\begin{array}{ccc}
4 & 1 & 2 \\
0 & 3 & 2 \\
1 & -2 & 3
\end{array}\right]=\left[\begin{array}{ccc}
0 & 0 & -3 \\
9 & -1 & 5 \\
2 & 1 & 1
\end{array}\right]
$$
From eqs. (iii) \& (iv) L.H.S. = R.H.S.
$$
\Rightarrow \mathrm{A}+(\mathrm{B}-\mathrm{C})=(\mathrm{A}+\mathrm{B})-\mathrm{C} .
$$
$$
\mathrm{B}-\mathrm{C}=\left[\begin{array}{ccc}
-1 & -2 & 0 \\
4 & -1 & 3 \\
1 & 2 & 0
\end{array}\right]
$$
L.H.S. A $+(\mathrm{B}-\mathrm{C})$ putting the value of $(\mathrm{B}-\mathrm{C})$ from (ii)
$$
\left[\begin{array}{ccc}
1 & 2 & -3 \\
5 & 0 & 2 \\
1 & -1 & 1
\end{array}\right]+\left[\begin{array}{ccc}
-1 & -2 & 0 \\
4 & -1 & 3 \\
1 & 2 & 0
\end{array}\right]=\left[\begin{array}{ccc}
0 & 0 & -3 \\
9 & -1 & 5 \\
2 & 1 & 1
\end{array}\right] \ldots
$$
R.H.S : $(A+B)-C$ putting the value of $(A+B)$ from (i)
$$
\left[\begin{array}{ccc}
4 & 1 & -1 \\
9 & 2 & 7 \\
3 & -1 & 4
\end{array}\right]-\left[\begin{array}{ccc}
4 & 1 & 2 \\
0 & 3 & 2 \\
1 & -2 & 3
\end{array}\right]=\left[\begin{array}{ccc}
0 & 0 & -3 \\
9 & -1 & 5 \\
2 & 1 & 1
\end{array}\right]
$$
From eqs. (iii) \& (iv) L.H.S. = R.H.S.
$$
\Rightarrow \mathrm{A}+(\mathrm{B}-\mathrm{C})=(\mathrm{A}+\mathrm{B})-\mathrm{C} .
$$
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