Matrices & Determinants
Matrix power relation from A²B = BA
MJAT_TS8_P1
Grade 12
Question:
Let $A$ and $B$ be $3\times 3$ invertible matrices such that $A^2B=BA$ and $P=A^{-1}$. Then the value of $P^3B^3A^3$ is:
A) $A^{24}B^3$
B) $A^{21}B^3$
C) $A^{18}B^3$
D) $A^{15}B^3$
Step-by-Step Solution
Key Concept: From $A^2B=BA$: post-multiply by $A$: $A^2BA=BA^2$. So $BA=A^2B\Rightarrow BA^n=A^{2n}B$. Also pre-multiply original by $B$: $BA^2B=B^2A$.
Answer: **B** ($A^{21}B^3$).
Correct Answer: B