Question

If $A$ , $B$ are two non-singular matrices of order $3$ and $I$ is an identity matrix of order $3$ such that $AA^{T}=5Ι$ and $3A^{- 1}=2A^{T}-Aadj\left(4 B\right)$ , 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)

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