Question

Find the matrix A such that `A=[[1,2,3],[4,5,6]]=`  `[[-7,-8,-9],[2,4,6]]`

Answer 1

\[\left( ii \right) Let A = \begin{bmatrix}w & x \\ y & z\end{bmatrix}\]
\[ \Rightarrow A\begin{bmatrix}1 & 2 & 3 \\ 4 & 5 & 6\end{bmatrix} = \begin{bmatrix}- 7 & - 8 & - 9 \\ 2 & 4 & 6\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}w & x \\ y & z\end{bmatrix}\begin{bmatrix}1 & 2 & 3 \\ 4 & 5 & 6\end{bmatrix} = \begin{bmatrix}- 7 & - 8 & - 9 \\ 2 & 4 & 6\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}w + 4x & 2w + 5x & 3w + 6x \\ y + 4z & 2y + 5z & 3y + 6z\end{bmatrix} = \begin{bmatrix}- 7 & - 8 & - 9 \\ 2 & 4 & 6\end{bmatrix}\]
\[\]
The correspnding elements of two equal matrices are equal .
\[ \therefore 3w + 6x = - 9 . . . \left( 1 \right) \]
\[ y + 4z = 2 \]
\[y = 2 - 4z . . . \left( 2 \right) \]
\[w + 4x = - 7 \]
\[ \Rightarrow w = - 7 - 4x . . . \left( 3 \right) \]
\[2y + 5z = 4 . . . \left( 4 \right) \]
putting the value of w   in eq.(1), we  get
\[3\left( - 7 - 4x \right) + 6x = - 9\]
\[ \Rightarrow - 21 - 12x + 6x = - 9\]
\[ \Rightarrow - 6x = 12\]
\[ \Rightarrow x = - 2\]
putting the value of x  in eq.(3), we  get
\[w = - 7 - 4\left( - 2 \right) \]
\[ \Rightarrow w = - 7 + 8\]
\[ \Rightarrow w = 1\]
putting the value of y   in eq.(4), we  get
\[2\left( 2 - 4z \right) + 5z = 4\]
\[ \Rightarrow 4 - 8z + 5z = 4\]
\[ \Rightarrow 4 - 3z = 4\]
\[ \Rightarrow - 3z = 0\]
\[ \Rightarrow z = 0\]
putting the value of  z in eq.(2), we  get
\[ y = 2 - 4\left( 0 \right) \]
\[ \Rightarrow y = 2\]
\[ \therefore A = \begin{bmatrix}1 & - 2 \\ 2 & 0\end{bmatrix}\]