Matrices & Determinants
Matrices And Determinants
nta_abhyas_2025
Grade None
Question:
If $A, B$ are two non-singular matrices of order $3$ and $I$ is an identity matrix of order $3$ such that $AA^T = 5I$ and $3A^{-1} = 2A^T$ - $\operatorname{adj}(AH)$, then $|B^n|$ is equal to
-\frac{125}{p^{2n}}
\frac{125}{p^{2n}}
-\frac{27}{p^{2n}}
-\frac{125}{p^{2n}}
Step-by-Step Solution
Key Concept: Properties of determinants: $|AA^T| = |A|^2$ and relationships between adjugate and determinant matrices
Given $|AA^T| = |B| \Rightarrow |A|^2 = b^3 - (1)$. Then $3A^T = 2AA^T - \text{adj}(4B)$, which upon multiplication by $A$ on both sides yields $3I = 2AA^T - \text{adj}(4B)$. Simplifying gives $|A|^2 |B|^2 = \frac{1}{27}$, so $|B|^2 = \frac{1}{27}$.
Correct Answer: 1