If A = [ ll 1 & 5 7 & 8 9 & 5 ], B = [ cc 2 & 4 1 & 5 -8 & 6 ], C… - Vidyadip

Question

If $A =\left[\begin{array}{ll}1 & 5 \\ 7 & 8 \\ 9 & 5\end{array}\right], B =\left[\begin{array}{cc}2 & 4 \\ 1 & 5 \\ -8 & 6\end{array}\right], C =\left[\begin{array}{cc}-2 & 3 \\ 1 & -5 \\ 7 & 8\end{array}\right]$ then show that $( A + B )+ C = A +( B + C )$.

✓

Answer

$
A+B=\left[\begin{array}{ll}
1 & 5 \\
7 & 8 \\
9 & 5
\end{array}\right)+\left(\begin{array}{rr}
2 & 4 \\
1 & 5 \\
-8 & 6
\end{array}\right)
$
$
=\left[\begin{array}{ll}
1+2 & 5+4 \\
7+1 & 8+5 \\
9-8 & 5+6
\end{array}\right]=\left[\begin{array}{rr}
3 & 9 \\
8 & 13 \\
1 & 11
\end{array}\right]
$
$
\begin{aligned}
& \therefore(A+B)+C=\left(\begin{array}{lr}
3 & 9 \\
8 & 13 \\
1 & 11
\end{array}\right)+\left[\begin{array}{rr}
-2 & 3 \\
1 & -5 \\
7 & 8
\end{array}\right) \\
& =\left(\begin{array}{rr}
3-2 & 9+3 \\
8+1 & 13-5 \\
1+7 & 11+8
\end{array}\right)=\left[\begin{array}{rr}
1 & 12 \\
9 & 8 \\
8 & 19
\end{array}\right) \ldots \ldots \ldots(1)
\end{aligned}
$
Also, $B + C =\left[\begin{array}{rr}2 & 4 \\ 1 & 5 \\ -8 & 6\end{array}\right]+\left[\begin{array}{rr}-2 & 3 \\ 1 & -5 \\ 7 & 8\end{array}\right]$
$
=\left(\begin{array}{rr}
2-2 & 4+3 \\
1+1 & 5-5 \\
-8+7 & 6+8
\end{array}\right)=\left(\begin{array}{rr}
0 & 7 \\
2 & 0 \\
-1 & 14
\end{array}\right)
$
$
\begin{aligned}
& \therefore A+(B+C)=\left(\begin{array}{ll}
1 & 5 \\
7 & 8 \\
9 & 5
\end{array}\right)+\left[\begin{array}{rr}
0 & 7 \\
2 & 0 \\
-1 & 14
\end{array}\right] \\
& =\left(\begin{array}{rr}
1+0 & 5+7 \\
7+2 & 8+0 \\
9-1 & 5+14
\end{array}\right)=\left(\begin{array}{rr}
1 & 12 \\
9 & 8 \\
8 & 19
\end{array}\right) \quad \ldots \ldots \ldots
\end{aligned}
$
From (1) and (2),
(A + B) + C = A + (B + C).

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