If \(\mathrm{A} = \begin{bmatrix} 2x & 0 \\ x & x \end{bmatrix}\) and \(\mathrm{A}^{-1} = \begin{bmatrix} 1 & 0 \\ -1 & 2 \end{bmatrix}\), then the value of x is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{1}{2}\)
A matrix multiplied by its inverse gives the identity matrix: \(AA^{-1} = I\).
\( \displaystyle AA^{-1} = \begin{bmatrix} 2x & 0 \\ x & x \end{bmatrix}\begin{bmatrix} 1 & 0 \\ -1 & 2 \end{bmatrix} \)
\( \displaystyle = \begin{bmatrix} 2x + 0 & 0 + 0 \\ x - x & 0 + 2x \end{bmatrix} \)
\( \displaystyle = \begin{bmatrix} 2x & 0 \\ 0 & 2x \end{bmatrix} \)
Equating this to \(I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\):
\(2x = 1\)
\( \displaystyle x = \frac{1}{2} \)