Matrices & Determinants
Matrix Power Formula — Solving Linear System
nta_pyq_2026_jan
Grade 12
Question:
For the matrices $A=\begin{bmatrix}3&-4\\1&-1\end{bmatrix}$ and $B=\begin{bmatrix}-29&49\\-13&18\end{bmatrix}$, if $(A^{15}+B)\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}$, then among the following which one is true?
$x=5,y=7$
$x=18,y=11$
$x=11,y=2$
$x=16,y=3$
Step-by-Step Solution
Key Concept: $A^n=\begin{bmatrix}2n+1&-4n\\n&-2n+1\end{bmatrix}$. So $A^{15}=\begin{bmatrix}31&-60\\15&-29\end{bmatrix}$. $A^{15}+B=\begin{bmatrix}2&-11\\2&-11\end{bmatrix}$.
$x=11$, $y=2$.
Correct Answer: 3