If $A$ and $B$ are square matrices of order 3 such that $4A^T = 3B$ and $2AB^T = 3A^T B$, then the value of $\frac{|A|^2}{|B|}$ is equal to
Step-by-Step Solution
Key Concept: Apply the determinant property $|kA| = k^n|A|$ for $n\times n$ matrices and use given matrix relations to form solvable equations.
From $4A^T = 3B$, taking determinant on both sides: $|4A^T| = |3B|$, so $4^3|A^T| = 3^3|B|$, giving $64|A| = 27|B|$. Also from $2AB^{-1} = 3A^TB$, we have $2|A||B|^{-1} = 3|A^T||B|$, so $2|A|/|B| = 3|A||B|$, which gives $2|A| = 3|A||B|^2$. Solving the system of equations yields $|B| = 8$.
Correct Answer: 8