If A= [ cc 2 & -1 3 & -2 4 & 1 ] and B= [ ccc 0 & 3 & -4 2 & -1 & 1 ], verify that (i) (A B)^ =B^ A^ (ii) (B A)^ =A^ B^

& A= ( rr 2 & -1 3 & -2 4 & 1 ), B = ( rrr 0 & 3 & -4 2 & -1 & 1 ) & A ^ T = ( rrr 2 & 3 & 4 -1 & -2 & 1 ), B ^ T = ( rr 0 & 2 3 & -1 -4 & 1 ) (2-1) (i) AB = ( rr 2 & -1 3 & -2 4 & 1 ) ( rrr 0 & 3 & -4 2 & -1 & 1 ) & = ( rrr 0-2 & 6+1 & -8-1 0-4 & 9+2 & -12-2 0+2 & 12-1 & -16+1 )= ( rrr -2 & 7 & -9 -4 & 11 & -14 2 & 11 & -15 ) & (A B)^ T = ( rrr -2 & -4 & 2 7 & 11 & 11 -9 & -14 & -15 ) & = ( rrr 0-2 & 0-4 & 0+2 6+1 & 9+2 & 12-1 -8-1 & -12-2 & -16+1 ) & = ( rrr -2 & -4 & 2 7 & 11 & 11 -9 & -14 & -15 ) From (1) and (2), ( AB )^ T = B ^ T A ^ T . (ii) BA = ( rrr 0 & 3 & -4 2 & -1 & 1 ) ( rr 2 & -1 3 & -2 4 & 1 ) & = ( rr 0+9-16 & 0-6-4 4-3+4 & -2+2+1 )= ( rr -7 & -10 5 & 1 ) & ( BA )^ T = ( rr -7 & 5 -10 & 1 ) A ^ T B ^ T & = ( rrr 2 & 3 & 4 -1 & -2 & 1 ) ( rr 0 & 2 3 & -1 -4 & 1 ) & = ( rr 0+9-16 & 4-3+4 0-6-4 & -2+2+1 ) & = ( rr -7 & 5 -10 & 1 ) From (1) and (2), (BA) ^ T = A ^ T B ^ T .

If A= [ cc 2 & -1 3 & -2 4 & 1 ] and B= [ ccc 0 & 3 & -4 2 & -1 &…

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Question

If $A=\left[\begin{array}{cc}2 & -1 \\ 3 & -2 \\ 4 & 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}0 & 3 & -4 \\ 2 & -1 & 1\end{array}\right]$, verify that
(i) $(A B)^{\top}=B^{\top} A^{\top}$
(ii) $(B A)^{\top}=A^{\top} B^{\top}$

✓

Answer

$
\begin{aligned}
& A=\left(\begin{array}{rr}
2 & -1 \\
3 & -2 \\
4 & 1
\end{array}\right), B =\left(\begin{array}{rrr}
0 & 3 & -4 \\
2 & -1 & 1
\end{array}\right) \\
& \therefore A ^{ T }=\left(\begin{array}{rrr}
2 & 3 & 4 \\
-1 & -2 & 1
\end{array}\right), B ^{ T }=\left(\begin{array}{rr}
0 & 2 \\
3 & -1 \\
-4 & 1
\end{array}\right)
\end{aligned}
$
$(2-1)$
(i) $AB =\left(\begin{array}{rr}2 & -1 \\ 3 & -2 \\ 4 & 1\end{array}\right)\left(\begin{array}{rrr}0 & 3 & -4 \\ 2 & -1 & 1\end{array}\right)$
$
\begin{aligned}
& =\left(\begin{array}{rrr}
0-2 & 6+1 & -8-1 \\
0-4 & 9+2 & -12-2 \\
0+2 & 12-1 & -16+1
\end{array}\right)=\left(\begin{array}{rrr}
-2 & 7 & -9 \\
-4 & 11 & -14 \\
2 & 11 & -15
\end{array}\right) \\
& \therefore(A B)^{ T }=\left(\begin{array}{rrr}
-2 & -4 & 2 \\
7 & 11 & 11 \\
-9 & -14 & -15
\end{array}\right)
\end{aligned}
$
$
\begin{aligned}
& =\left(\begin{array}{rrr}
0-2 & 0-4 & 0+2 \\
6+1 & 9+2 & 12-1 \\
-8-1 & -12-2 & -16+1
\end{array}\right) \\
& =\left(\begin{array}{rrr}
-2 & -4 & 2 \\
7 & 11 & 11 \\
-9 & -14 & -15
\end{array}\right)
\end{aligned}
$
From (1) and (2),
$
( AB )^{ T }= B ^{ T } A ^{ T } \text {. }
$
(ii) $BA =\left(\begin{array}{rrr}0 & 3 & -4 \\ 2 & -1 & 1\end{array}\right)\left(\begin{array}{rr}2 & -1 \\ 3 & -2 \\ 4 & 1\end{array}\right)$
$
\begin{aligned}
& \quad=\left(\begin{array}{rr}
0+9-16 & 0-6-4 \\
4-3+4 & -2+2+1
\end{array}\right)=\left(\begin{array}{rr}
-7 & -10 \\
5 & 1
\end{array}\right) \\
& \therefore( BA )^{ T }=\left(\begin{array}{rr}
-7 & 5 \\
-10 & 1
\end{array}\right)
\end{aligned}
$
$
\begin{aligned}
A ^{ T } B ^{ T } & =\left(\begin{array}{rrr}
2 & 3 & 4 \\
-1 & -2 & 1
\end{array}\right)\left(\begin{array}{rr}
0 & 2 \\
3 & -1 \\
-4 & 1
\end{array}\right) \\
& =\left(\begin{array}{rr}
0+9-16 & 4-3+4 \\
0-6-4 & -2+2+1
\end{array}\right) \\
& =\left(\begin{array}{rr}
-7 & 5 \\
-10 & 1
\end{array}\right)
\end{aligned}
$
From (1) and (2), (BA) ${ }^{ T }= A ^{ T } B ^{ T }$.

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