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Question
Find a 2 × 2 matrix A such that `A=[[1,-2],[1,4]]=6l_2`
Solution
let `A=[[w,x],[y,z]]`
\[\begin{bmatrix}w & x \\ y & z\end{bmatrix}\begin{bmatrix}1 & - 2 \\ 1 & 4\end{bmatrix} = 6 I_2 \]
\[ \Rightarrow \begin{bmatrix}w + x & - 2w + 4x \\ y + z & - 2y + 4z\end{bmatrix} = 6\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}w + x & - 2w + 4x \\ y + z & - 2y + 4z\end{bmatrix} = \begin{bmatrix}6 & 0 \\ 0 & 6\end{bmatrix}\]
The corresponding elements of two equal matrices are equal .
\[ \therefore w + x = 6 \]
\[ \Rightarrow w = 6 - x . . . \left( 1 \right) \]
\[ - 2w + 4x = 0 . . . \left( 2 \right) \]
Putting the value of w in eq .(2) we get
\[ - 2\left( 6 - x \right) + 4x = 0\]
\[ \Rightarrow - 12 + 2x + 4x = 0\]
\[ \Rightarrow - 12 + 6x = 0\]
\[ \Rightarrow 6x = 12\]
\[ \Rightarrow x = 2\]
putting the value of x in eq.(1) we get
\[w = 6 - 2\]
\[ \Rightarrow w = 4\]
\[Now, \]
\[y + z = 0\]
\[ \Rightarrow y = - z . . . \left( 3 \right) \]
\[ - 2y + 4z = 6 . . . \left( 4 \right) \]
Putting the value of y in eq .( 4), we get
\[ - 2\left( - z \right) + 4z = 6\]
\[ \Rightarrow 2z + 4z = 6\]
\[ \Rightarrow 6z = 6\]
\[ \Rightarrow z = 1\]
Putting the value of z in eq .( 3 ), we get
\[y = - 1\]
\[ \therefore A = \begin{bmatrix}4 & 2 \\ - 1 & 1\end{bmatrix}\]
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