If A= [ cc 2 & -3 5 & -4 -6 & 1 ], B= [ cc -1 & 2 2 & 2 0 & 3 ] and C= [ cc 4 & 3 -1 & 4 -2 & 1 ] show that (i) A + B = B + A (ii) (A+B)+C=A+(B+C)

& A+B= ( rr 2 & -3 5 & -4 -6 & 1 )+ [ rr -1 & 2 2 & 2 0 & 3 ] & = ( rr 2-1 & -3+2 5+2 & -4+2 -6+0 & 1+3 )= [ rr 1 & -1 7 & -2 -6 & 4 ] & B + A = [ rr -1 & 2 2 & 2 0 & 3 ]+ [ rr 2 & -3 5 & -4 -6 & 1 ] & = [ rr -1+2 & 2-3 2+5 & 2-4 0-6 & 3+1 ]= [ rr 1 & -1 7 & -2 -6 & 4 ] & A+B= ( rr 2 & -3 5 & -4 -6 & 1 )+ ( rr -1 & 2 2 & 2 0 & 3 ) & = [ rr 2-1 & -3+2 5+2 & -4+2 -6+0 & 1+3 ]= ( rr 1 & -1 7 & -2 -6 & 4 ) & (A+B)+C= ( rr 1 & -1 7 & -2 -6 & 4 )+ ( rr 4 & 3 -1 & 4 -2 & 1 ) & = [ rr 1+4 & -1+3 7-1 & -2+4 -6-2 & 4+1 ]= [ rr 5 & 2 6 & 2 -8 & 5 ] & Also, B + C = ( rr -1 & 2 2 & 2 0 & 3 )+ ( rr 4 & 3 -1 & 4 -2 & 1 ) & = [ rr -1+4 & 2+3 2-1 & 2+4 0-2 & 3+1 ]= [ rr 3 & 5 1 & 6 -2 & 4 ] A & +(B+C)= [ rr 2 & -3 5 & -4 -6 & 1 ]+ [ rr 3 & 5 1 & 6 -2 & 4 ] & = [ rr 2+3 & -3+5 5+1 & -4+6 -6-2 & 1+4 ]= [ rr 5 & 2 6 & 2 -8 & 5 ] From (1) and (2), we get (A+B)+C=A+(B+C)

If A= [ cc 2 & -3 5 & -4 -6 & 1 ], B= [ cc -1 & 2 2 & 2 0 & 3 ] a…

Download App

Question

If $A=\left[\begin{array}{cc}2 & -3 \\ 5 & -4 \\ -6 & 1\end{array}\right], B=\left[\begin{array}{cc}-1 & 2 \\ 2 & 2 \\ 0 & 3\end{array}\right]$ and $C=\left[\begin{array}{cc}4 & 3 \\ -1 & 4 \\ -2 & 1\end{array}\right]$ show that
(i) $A + B = B + A$
(ii) $(A+B)+C=A+(B+C)$

✓

Answer

$
\begin{aligned}
& A+B=\left(\begin{array}{rr}
2 & -3 \\
5 & -4 \\
-6 & 1
\end{array}\right)+\left[\begin{array}{rr}
-1 & 2 \\
2 & 2 \\
0 & 3
\end{array}\right] \\
& =\left(\begin{array}{rr}
2-1 & -3+2 \\
5+2 & -4+2 \\
-6+0 & 1+3
\end{array}\right)=\left[\begin{array}{rr}
1 & -1 \\
7 & -2 \\
-6 & 4
\end{array}\right] \\
& B + A =\left[\begin{array}{rr}
-1 & 2 \\
2 & 2 \\
0 & 3
\end{array}\right]+\left[\begin{array}{rr}
2 & -3 \\
5 & -4 \\
-6 & 1
\end{array}\right] \\
& =\left[\begin{array}{rr}
-1+2 & 2-3 \\
2+5 & 2-4 \\
0-6 & 3+1
\end{array}\right]=\left[\begin{array}{rr}
1 & -1 \\
7 & -2 \\
-6 & 4
\end{array}\right] \\
& A+B=\left(\begin{array}{rr}
2 & -3 \\
5 & -4 \\
-6 & 1
\end{array}\right)+\left(\begin{array}{rr}
-1 & 2 \\
2 & 2 \\
0 & 3
\end{array}\right) \\
& =\left[\begin{array}{rr}
2-1 & -3+2 \\
5+2 & -4+2 \\
-6+0 & 1+3
\end{array}\right]=\left(\begin{array}{rr}
1 & -1 \\
7 & -2 \\
-6 & 4
\end{array}\right) \\
& \therefore(A+B)+C=\left(\begin{array}{rr}
1 & -1 \\
7 & -2 \\
-6 & 4
\end{array}\right)+\left(\begin{array}{rr}
4 & 3 \\
-1 & 4 \\
-2 & 1
\end{array}\right) \\
& =\left[\begin{array}{rr}
1+4 & -1+3 \\
7-1 & -2+4 \\
-6-2 & 4+1
\end{array}\right]=\left[\begin{array}{rr}
5 & 2 \\
6 & 2 \\
-8 & 5
\end{array}\right] \\
&
\end{aligned}
$
Also, $B + C =\left(\begin{array}{rr}-1 & 2 \\ 2 & 2 \\ 0 & 3\end{array}\right)+\left(\begin{array}{rr}4 & 3 \\ -1 & 4 \\ -2 & 1\end{array}\right)$
$
\begin{aligned}
& =\left[\begin{array}{rr}
-1+4 & 2+3 \\
2-1 & 2+4 \\
0-2 & 3+1
\end{array}\right]=\left[\begin{array}{rr}
3 & 5 \\
1 & 6 \\
-2 & 4
\end{array}\right] \\
\therefore A & +(B+C)=\left[\begin{array}{rr}
2 & -3 \\
5 & -4 \\
-6 & 1
\end{array}\right]+\left[\begin{array}{rr}
3 & 5 \\
1 & 6 \\
-2 & 4
\end{array}\right] \\
& =\left[\begin{array}{rr}
2+3 & -3+5 \\
5+1 & -4+6 \\
-6-2 & 1+4
\end{array}\right]=\left[\begin{array}{rr}
5 & 2 \\
6 & 2 \\
-8 & 5
\end{array}\right]
\end{aligned}
$
From (1) and (2), we get
$
(A+B)+C=A+(B+C)
$