Matrices & Determinants
A³=A Condition — Finding n
nta_pyq_2023_apr
Grade 12
Question:
Let $A=\begin{pmatrix}0&1&2\\a&0&3\\1&c&0\end{pmatrix}$, $a,c\in\mathbb{R}$. If $A^3=A$ and the positive value of $a$ belongs to $(n-1,n]$, $n\in\mathbb{N}$, then $n$ is equal to ____.
Step-by-Step Solution
Key Concept: Compute $A^3$ and set $A^3=A$. This gives conditions: $c=-\frac{3}{2a}$ and $2a^2+2a-9=0$.
Positive root $\approx1.4\in(1,2]$. $n=2$.
Correct Answer: 2