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Question
Find the matrix A such that `[[2,-1],[1,0],[-3,-4]]A` `=[[-1,-8,-10],[1,-2,-5],[9,22,15]]`
Solution
\[\left( v \right) \begin{bmatrix}2 & - 1 \\ 1 & 0 \\ - 3 & 4\end{bmatrix}A = \begin{bmatrix}- 1 & - 8 & - 10 \\ 1 & - 2 & - 5 \\ 9 & 22 & 15\end{bmatrix}\]
\[Let A = \begin{bmatrix}x & y & z \\ a & b & c\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}2 & - 1 \\ 1 & 0 \\ - 3 & 4\end{bmatrix}\begin{bmatrix}x & y & z \\ a & b & c\end{bmatrix} = \begin{bmatrix}- 1 & - 8 & - 10 \\ 1 & - 2 & - 5 \\ 9 & 22 & 15\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}2x - a & 2y - b & 2z - c \\ x & y & z \\ - 3x + 4a & - 3y + 4b & - 3z + 4c\end{bmatrix} = \begin{bmatrix}- 1 & - 8 & - 10 \\ 1 & - 2 & - 5 \\ 9 & 22 & 15\end{bmatrix}\]
By comparing the elements of second row, we get
\[x = 1, y = - 2, z = - 5\]
By comparing the elements of first row, we get
\[2x - a = - 1\]
\[ \Rightarrow 2 - a = - 1\]
\[ \Rightarrow a = 3\]
\[Also, \]
\[2y - b = - 8\]
\[ \Rightarrow - 4 - b = - 8\]
\[ \Rightarrow b = 4\]
\[And, \]
\[2z - c = - 10\]
\[ \Rightarrow - 10 - c = - 10\]
\[ \Rightarrow c = 0\]
\[ \therefore A = \begin{bmatrix}1 & - 2 & - 5 \\ 3 & 4 & 0\end{bmatrix}\]
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