Question

Let \[A = \begin{bmatrix}1 & 2 \\ 3 & - 5\end{bmatrix}\text{ and }B = \begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix}\] and X be a matrix such that A = BX, then X is equal to _____________ .

Options

• \[\frac{1}{2}\begin{bmatrix}2 & 4 \\ 3 & - 5\end{bmatrix}\]

• \[\frac{1}{2}\begin{bmatrix}- 2 & 4 \\ 3 & 5\end{bmatrix}\]

• \[\begin{bmatrix}2 & 4 \\ 3 & - 5\end{bmatrix}\]

• none of these

Answer 1

\[\frac{1}{2}\begin{bmatrix}2 & 4 \\ 3 & - 5\end{bmatrix}\]
\[A = BX\]

\[ \Rightarrow B^{- 1} A = B^{- 1} BX\]

\[ \Rightarrow B^{- 1} A = IX\]

\[ \Rightarrow X = B^{- 1} A . . . \left( 1 \right)\]

Now,

\[B = \begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix}\]

\[adjB = \begin{bmatrix}2 & 0 \\ 0 & 1\end{bmatrix}\]

\[\left| B \right| = 2\]

\[ \therefore B^{- 1} = \frac{1}{\left| B \right|}adjB = \frac{1}{2}\begin{bmatrix}2 & 0 \\ 0 & 1\end{bmatrix}\]

On putting the value of B^-1 in eq. (1), we get

\[X = \frac{1}{2}\begin{bmatrix}2 & 0 \\ 0 & 1\end{bmatrix}\begin{bmatrix}1 & 2 \\ 3 & - 5\end{bmatrix}\]

\[ \Rightarrow X = \frac{1}{2}\begin{bmatrix}2 & 4 \\ 3 & - 5\end{bmatrix}\]