Question

If \[\begin{bmatrix}1 & 0 \\ y & 5\end{bmatrix} + 2\begin{bmatrix}x & 0 \\ 1 & - 2\end{bmatrix}\]  = I, where I is 2 × 2 unit matrix. Find x and y.

 

Answer 1

\[Given: \hspace{0.167em} \begin{bmatrix}1 & 0 \\ y & 5\end{bmatrix} + 2\begin{bmatrix}x & 0 \\ 1 & - 2\end{bmatrix} = I\] 

\[ \Rightarrow \begin{bmatrix}1 & 0 \\ y & 5\end{bmatrix} + \begin{bmatrix}2x & 0 \\ 2 & - 4\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\] 

\[ \Rightarrow \begin{bmatrix}1 + 2x & 0 + 0 \\ y + 2 & 5 - 4\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\] 

\[ \Rightarrow \begin{bmatrix}1 + 2x & 0 \\ y + 2 & 1\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\] 

\[ \therefore 1 + 2x = \text{1 and y + 2} = 0\] 

\[ \Rightarrow 2x =\text{ 1 - 1 and y} = - 2\] 

\[ \Rightarrow 2x = 0\] 

\[ \Rightarrow x = \frac{0}{2} = 0\]