If \(\begin{bmatrix}3 & 2\\ 4 & 3\end{bmatrix}X = \begin{bmatrix}1 & 0\\ 2 & -1\end{bmatrix}\), then the matrix X is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\begin{bmatrix}-1 & 2\\ 2 & -3\end{bmatrix}\)
Let \(A = \begin{bmatrix}3 & 2\\ 4 & 3\end{bmatrix}\) and \(B = \begin{bmatrix}1 & 0\\ 2 & -1\end{bmatrix}\). The equation is AX = B, so multiplying by \(A^{-1}\) on the left gives \(X = A^{-1}B\).
\( |A| = 3 \times 3 - 2 \times 4 = 1 \)
\( \displaystyle A^{-1} = \begin{bmatrix}3 & -2\\ -4 & 3\end{bmatrix} \)
\( \displaystyle X = \begin{bmatrix}3 & -2\\ -4 & 3\end{bmatrix}\begin{bmatrix}1 & 0\\ 2 & -1\end{bmatrix} \)
\( \displaystyle = \begin{bmatrix}3 - 4 & 0 + 2\\ -4 + 6 & 0 - 3\end{bmatrix} = \begin{bmatrix}-1 & 2\\ 2 & -3\end{bmatrix} \)
Why the other options are incorrect
Option (1) \(\begin{bmatrix}3 & -2\\ 10 & -7\end{bmatrix}\) — Incorrect. This is \(BA^{-1}\); matrix products depend on order, and AX = B needs \(A^{-1}\) on the left.
Option (2) \(\begin{bmatrix}7 & -2\\ 10 & -3\end{bmatrix}\) — Incorrect. This is AB, which uses A instead of \(A^{-1}\).
Option (4) \(\begin{bmatrix}1 & -2\\ -2 & 3\end{bmatrix}\) — Incorrect. Every sign is reversed; with this matrix the (1, 1) entry of AX is \(3 - 4 = -1\), not 1.