Matrices & Determinants
Matrix Reconstruction and System Solving
nta_pyq_2024_jan
Grade 12
Question:
Let $A$ be a $3\times3$ real matrix such that $A\begin{pmatrix}1\\0\\1\end{pmatrix}=2\begin{pmatrix}1\\0\\1\end{pmatrix}$, $A\begin{pmatrix}-1\\0\\1\end{pmatrix}=4\begin{pmatrix}-1\\0\\1\end{pmatrix}$, $A\begin{pmatrix}0\\1\\0\end{pmatrix}=2\begin{pmatrix}0\\1\\0\end{pmatrix}$. Then, the system $(A-3I)\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}1\\2\\3\end{pmatrix}$ has
unique solution
exactly two solutions
no solution
infinitely many solutions
Step-by-Step Solution
Key Concept: Use the given matrix-vector equations to reconstruct $A$ by solving for the entries. Then compute $A-3I$ and solve the system — check whether $(A-3I)$ is invertible.
From given: $x_1=3x, x_2=0, x_3=-1$; $y_1=0, y_2=2, y_3=0$; $z_1=-1, z_2=0, z_3=3$. $A=\begin{bmatrix}3&0&-1\\0&2&0\\-1&0&3\end{bmatrix}$. $A-3I=\begin{bmatrix}0&0&-1\\0&-1&0\\-1&0&0\end{bmatrix}$. $\det(A-3I)=-1\neq0$. Unique solution: $z=-1, y=-2, x=-3$.
Correct Answer: 1