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Transpose of a Matrix

For the matri...

Question

For the matrices $A$ and $B$ , verify that ${(A B)}^{'} = B^{'} A^{'}$ where
(i) $A = ⎡ ⎢ ⎣ \begin{matrix} 1 - 4 3 \end{matrix} ⎤ ⎥ ⎦, B = [\begin{matrix} - 1 & 2 & 1 \end{matrix}]$
(ii) $A = ⎡ ⎢ ⎣ \begin{matrix} 012 \end{matrix} ⎤ ⎥ ⎦, B = [\begin{matrix} 1 & 5 & 7 \end{matrix}]$

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Solution

(i) $A B = ⎡ ⎢ ⎣ \begin{matrix} 1 - 4 3 \end{matrix} ⎤ ⎥ ⎦ [\begin{matrix} - 1 & 2 & 1 \end{matrix}] = ⎡ ⎢ ⎣ \begin{matrix} - 1 4 - 3 \end{matrix} \begin{matrix} 2 - 8 6 \end{matrix} \begin{matrix} 1 - 4 3 \end{matrix} ⎤ ⎥ ⎦$
$∴ {(A B)}^{'} = ⎡ ⎢ ⎣ \begin{matrix} - 1 21 \end{matrix} \begin{matrix} 4 - 8 - 4 \end{matrix} \begin{matrix} - 3 63 \end{matrix} ⎤ ⎥ ⎦$

Now, $A^{'} = [\begin{matrix} 1 & - 4 & 3 \end{matrix}], B^{'} = ⎡ ⎢ ⎣ \begin{matrix} - 1 21 \end{matrix} ⎤ ⎥ ⎦$
$∴ B^{'} A^{'} = ⎡ ⎢ ⎣ \begin{matrix} - 1 21 \end{matrix} ⎤ ⎥ ⎦ [\begin{matrix} 1 & - 4 & 3 \end{matrix}] = ⎡ ⎢ ⎣ \begin{matrix} - 1 21 \end{matrix} \begin{matrix} 4 - 8 - 4 \end{matrix} \begin{matrix} - 3 63 \end{matrix} ⎤ ⎥ ⎦$

Hence, we have verified that ${(A B)}^{'} = B^{'} A^{'}$

(ii) $A B = ⎡ ⎢ ⎣ \begin{matrix} 012 \end{matrix} ⎤ ⎥ ⎦ [\begin{matrix} 1 & 5 & 7 \end{matrix}] = ⎡ ⎢ ⎣ \begin{matrix} 012 \end{matrix} \begin{matrix} 0510 \end{matrix} \begin{matrix} 0714 \end{matrix} ⎤ ⎥ ⎦$
$∴ {(A B)}^{'} = ⎡ ⎢ ⎣ \begin{matrix} 000 \end{matrix} \begin{matrix} 157 \end{matrix} \begin{matrix} 21014 \end{matrix} ⎤ ⎥ ⎦$

Now, $A^{'} = [\begin{matrix} 0 & 1 & 2 \end{matrix}], B^{'} = ⎡ ⎢ ⎣ \begin{matrix} 157 \end{matrix} ⎤ ⎥ ⎦$
$∴ B^{'} A^{'} = ⎡ ⎢ ⎣ \begin{matrix} 157 \end{matrix} ⎤ ⎥ ⎦ [\begin{matrix} 0 & 1 & 2 \end{matrix}] = ⎡ ⎢ ⎣ \begin{matrix} 000 \end{matrix} \begin{matrix} 157 \end{matrix} \begin{matrix} 21014 \end{matrix} ⎤ ⎥ ⎦$

Hence, we have proved that ${(A B)}^{'} = B^{'} A^{'}$

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For the matrices A and B, verify that (AB)′ = where

(i)

(ii)

Q. For the matrices A and B , verify that ( AB )′ = where (i) (ii)

Q. (i) For two matrices A and B,

A = [\begin{matrix} 2 & 1 & 3 \\ 4 & 1 & 0 \end{matrix}], B = [\begin{matrix} 1 & - 1 \\ 0 & 2 \\ 5 & 0 \end{matrix}]

verify that
(AB)^T = B^T A^T.

(ii) For the matrices A and B, verify that (AB)^T = B^T A^T, where

A = [\begin{matrix} 1 & 3 \\ 2 & 4 \end{matrix}], B = [\begin{matrix} 1 & 4 \\ 2 & 5 \end{matrix}]

For the matrices A and B, verify that (AB)' =B'A', where
A= $⎡ ⎢ ⎣ \begin{matrix} 012 \end{matrix} ⎤ ⎥ ⎦$ , B =[1 5 7]

For the matrices A and B, verify that (AB)' =B'A', where $A = ⎡ ⎢ ⎣ \begin{matrix} 1 - 4 3 \end{matrix} ⎤ ⎥ ⎦, B = [- 121]$

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