Find the matrix A such that A= 1&2&3 4&5&6 = -7&-8&-9 2&4&6 11&10… - Vidyadip

Question

Find the matrix A such that
$\text{A}=\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}=\begin{bmatrix}-7&-8&-9\\2&4&6\\11&10&9\end{bmatrix}$

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Answer

Let $\text{A}=\begin{bmatrix}\text{x}&\text{a}\\\text{y}&\text{b}\\\text{z}&\text{c}\end{bmatrix}$
$\Rightarrow\begin{bmatrix}\text{x}&\text{a}\\\text{y}&\text{b}\\\text{z}&\text{c}\end{bmatrix}\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}=\begin{bmatrix}-7&-8&-9\\2&4&6\\11&10&9\end{bmatrix}$
$\Rightarrow\begin{bmatrix}\text{x}+4\text{a}&2\text{x}+5\text{a}&3\text{x}+6\text{a}\\\text{y}+4\text{b}&2\text{y}+5\text{b}&3\text{y}+6\text{b}\\\text{z}+4\text{c}&2\text{z}+5\text{c}&3\text{z}+6\text{c}\end{bmatrix}=\begin{bmatrix}-7&-8&-9\\2&4&6\\11&10&9\end{bmatrix}$
By comparing the corresponding elements, we get
x + 4a = -7 and 2x + 5a = -8
⇒ a = -2 and x = 1
Also,
y + 4b = 2 and 2y + 5b = 4
⇒ b = 0 and y = 2
And
z + 4c = 11 and 2z + 5b = 10
⇒ c = 4 and z = -5
$\therefore\ \text{A}=\begin{bmatrix}1&-2\\2&0\\-5&4\end{bmatrix}$

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