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Question
Find the matrix A such that `=[[1,2,3],[4,5,6]]=[[-7,-8,-9],[2,4,6],[11,10,9]]`
Solution
\[\left( vi \right) A\begin{bmatrix}1 & 2 & 3 \\ 4 & 5 & 6\end{bmatrix} = \begin{bmatrix}- 7 & - 8 & - 9 \\ 2 & 4 & 6 \\ 11 & 10 & 9\end{bmatrix}\]
\[Let A = \begin{bmatrix}x & a \\ y & b \\ z & c\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}x & a \\ y & b \\ z & c\end{bmatrix}\begin{bmatrix}1 & 2 & 3 \\ 4 & 5 & 6\end{bmatrix} = \begin{bmatrix}- 7 & - 8 & - 9 \\ 2 & 4 & 6 \\ 11 & 10 & 9\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}x + 4a & 2x + 5a & 3x + 6a \\ y + 4b & 2y + 5b & 3y + 6b \\ z + 4c & 2z + 5c & 3z + 6c\end{bmatrix} = \begin{bmatrix}- 7 & - 8 & - 9 \\ 2 & 4 & 6 \\ 11 & 10 & 9\end{bmatrix}\]
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 + 5c = 10
`⇒ c= 4` and `z=-5`
\[ \therefore A = \begin{bmatrix}1 & - 2 \\ 2 & 0 \\ - 5 & 4\end{bmatrix}\]
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