Question

If $A$ , $B$ are two non-singular matrices of order $3$ and $I$ is an identity matrix of order $3$ such that $A{A}^T=5I$ and $3A^{-1}=2A^T-Aadj\left(4B\right)$ , then ${\left|B\right|}^2$ is equal to (where $A^T$ and $adj\left(A\right)$ denote transpose and adjoint matrices of the matrix $A$ respectively)

• A. $\frac{7^3}{5^3\cdot 4^6}$
• B. $\frac{7^3}{5^3\cdot 4^3}$
• C. $\frac{7^4}{5^3\cdot 2^{12}}$
• D. $\frac{5^6}{7^3\cdot 2^{10}}$

Answer 1

Answer: (A)

$\left|A{A}^T\right|=\left|5I\right|\Rightarrow {\left|A\right|}^2=5^3$ ....(1)
$3A^{-1}=2A^T-Aadj\left(4B\right)$
Multiplying by $A$ on both sides, we get
$3I=2A{A}^T-A^{2}adj\left(4B\right)$
$\Rightarrow A^{2}adj\left(4B\right)=7I$
${\left|A\right|}^2\left|adj4B\right|=7^3$
$5^3{4}^6{\left|B\right|}^2=7^3$ (using (1))
${\left|B\right|}^2=\frac{7^3}{5^3\cdot 4^6}$