If A=pmatrix15!6!7! & & 5!

Question

If $A=\begin{pmatrix}\frac1{5!6!7!} & & \ 5! & 6! & 7!\ 6! & 7! & 8!\ 7! & 8! & 9!\end{pmatrix}$ (a $3	imes3$ matrix with the given scaled factorial entries), then $|\operatorname{adj}(\operatorname{adj}(2A))|$ is equal to

Accepted Answer

Row-reducing the matrix (factoring out $5!,6!,7!$ from the rows, then reducing) gives $|A|=2$.

For a $3	imes3$ matrix, $|\operatorname{adj}(\operatorname{adj}(2A))|=|2A|^{(3-1)^2}=|2A|^4=(2^3|A|)^4=(2^3\cdot2)^4=(2^4)^4=2^{16}$.

Solutions · 1