Matrices & Determinants
Matrices And Determinants
nta_abhyas_2025
Grade 12
Question:
A square matrix $A$ of order 3 satisfies $A^2 = I - 2A$, where $I$ is an identity matrix of order 3. If $A^n = 29A - 12I$, then the value of $n$ is equal to
Step-by-Step Solution
Key Concept: Use the given relation $P = I - 2A$ to iteratively compute successive powers of $A$ by reducing higher order terms using the relation.
We compute $A^1 = PA = (I - 2A)A = A - 2A^2 = A - 2(I - 2A) = 5A - 2I$. Then $A^2 = PA^1 = (I - 2A)(5A - 2I) = 5A^2 - 2I - 10A^2 + 4A = 5I - 12A - 5I = -12A - 5I$. Finally, $A^3 = A^1(A) = (5I - 12A)A = 5A - 12A^2 = 5A - 12(I - 2A) = 29A - 12I$.
Correct Answer: 5