if A'= 3&4 -1&2 0&1 ,and B= -1&2&1 1&2&3 ,then verify that 1. (A+B)'=A'+B' 2. (A-B)'=A'-B'

1. It is know that A = (A')' Therefore, we have: A= 3&-1&0 4&2&1 B'= -1&1 2&2 1&3 A + B= 3&-1&0 4&2&1 + -1&2&1 1&2&3 = 2&1&1 5&4&4 (A + B)'= 2&5 1&4 1&4 A'+B'= 3&4 -1&2 0&1 + -1&1 2&2 1&3 = 2&5 1&4 1&4 Thus, we have verify that (A + B)' = A' + B' 2. A -B= 3&-1&0 4&2&1 - -1&2&1 1&2&3 = 4&-3&-1 3&0&-2 (A - B)'= 4&3 -3&0 -1&-2 A' - B'= 3&4 -1&2 0&1 -1&1 2&2 1&3 = 4&3 -3&0 -1&-2 Thus, we have verify that (A - B)' = A' - B'

if A'= 3&4 -1&2 0&1 ,and B= -1&2&1 1&2&3 ,then verify that 1. (A+…

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Question

$\text{if} \ \text{A}'=\begin{bmatrix}3&4\\-1&2\\0&1\end{bmatrix},\text{and}\ \text{B}=\begin{bmatrix}-1&2&1\\1&2&3\end{bmatrix},\text{then verify that}$

$\text{(A+B)}'=\text{A}'+\text{B}'$
$\text{(A}-\text{B)}'=\text{A}'-\text{B}'$

✓

Answer

It is know that $ \text{A} = (\text{A}')'$

Therefore, we have:

$\text{A}=\begin{bmatrix}3&-1&0\\4&2&1\end{bmatrix} $

$ \text{B}'=\begin{bmatrix}-1&1\\2&2\\1&3\end{bmatrix}$

$\text{A + B}=\begin{bmatrix}3&-1&0\\4&2&1\end{bmatrix}+\begin{bmatrix}-1&2&1\\1&2&3\end{bmatrix}=\begin{bmatrix}2&1&1\\5&4&4\end{bmatrix}$

$\therefore\text{(A + B)}'=\begin{bmatrix}2&5\\1&4\\1&4\end{bmatrix}$

$\text{A}'+\text{B}'=\begin{bmatrix}3&4\\-1&2\\0&1\end{bmatrix}+\begin{bmatrix}-1&1\\2&2\\1&3\end{bmatrix}=\begin{bmatrix}2&5\\1&4\\1&4\end{bmatrix}$

Thus, we have verify that $ (\text{A} + \text{B})' = \text{A}' + \text{B}'$

$\text{A} -\text{B}=\begin{bmatrix}3&-1&0\\4&2&1\end{bmatrix}-\begin{bmatrix}-1&2&1\\1&2&3\end{bmatrix}=\begin{bmatrix}4&-3&-1\\3&0&-2\end{bmatrix}$

$\therefore\text{(A} - \text{B})'=\begin{bmatrix}4&3\\-3&0\\-1&-2\end{bmatrix}$

$\text{A}' - \text{B}'=\begin{bmatrix}3&4\\-1&2\\0&1\end{bmatrix}\begin{bmatrix}-1&1\\2&2\\1&3\end{bmatrix}=\begin{bmatrix}4&3\\-3&0\\-1&-2\end{bmatrix}$

Thus, we have verify that $ (\text{A} - \text{B})' = \text{A}' - \text{B}'$