Answer to Question #137167 in Linear Algebra for Lethu

Question

Suppose A is a 3x3 matrix such that det(A) =16.Find the value of det[(4(A^-1)^T]

Answer 1

Taking into account that $\det(M^{-1})=\frac{1}{\det (M)}$ and $\det(M^T)=\det (M)$ for any matrix $M$, we conclude that $\det ((A^{-1})^T)=\det (A^{-1})=\frac{1}{\det A}=\frac{1}{16}.$

Since $\det(kM)=k^3\det(M)$ for any $3\times 3$ matrix $M$,

$\det (4(A^{-1})^T)=4^3\det ((A^{-1})^T)=4^3\frac{1}{16}=4.$

Answer: $4.$