Let A and B be two square matrices of order 3 such that |A|=3 and |B|=2.

Question

Let A and B be two square matrices of order 3 such that $|A|=3$ and $|B|=2$. Then $|{A}^{T}A(adj(2A){)}^{-1}(adj(4B))(adj(AB){)}^{-1}{AA}^{T}|$ is equal to :

Accepted Answer

$\begin{gathered} |A|=3 \ |B|=2 \ |{A}^{T}||A||(adj(2A){)}^{-1}\parallel adj(4B)\parallel (adj(AB){)}^{-1})|A||{A}^{T}| \ 3\cdot 3\frac{1}{64\cdot 9}(64{)}^{2}\cdot 4\cdot \frac{1}{9\cdot 4}3\cdot 3 \ =64 \end{gathered}$

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