Matrices & Determinants
Matrix Power — Trace of A^13
nta_pyq_2024_apr
Grade 12
Question:
Let $A=\begin{bmatrix}2&-1\\1&1\end{bmatrix}$. If the sum of the diagonal elements of $A^{13}$ is $3^n$, then $n$ is equal to
Step-by-Step Solution
Key Concept: Compute powers of $A$ until a pattern emerges. $A^6=-27I$, so $A^{12}=(-27)^2I=729I=3^6I$. Then $A^{13}=A^{12}\cdot A=3^6 A$.
$A^6=-27I$, $A^{12}=3^6I$. Trace of $A^{13}=3^7$. $n=7$.
Correct Answer: 7