Mathematics - Matrices Question with Solution | TestHub

Let,$A={\left[a_{ii}\right]}_{3\times 3}$
$tr\left(A{A}^T\right)=3$
$a_{11}^2+a_{12}^2\mathrm{}+a_{13}^2+a_{21}^2+.......+a_{33}^2=3$
Possible cases
$\left.\begin{array}{cc}0,\mathrm{0,0},\mathrm{0,0},\mathrm{0,1},\mathrm{1,1}& \to 1\\ 0,\mathrm{0,0},\mathrm{0,0},0,-1,-1,-1& \to 1\\ 0,\mathrm{0,0},\mathrm{0,0},\mathrm{0,1},1,-1& \to 3\\ 0,\mathrm{0,0},\mathrm{0,0},0,-\mathrm{1,1},-1& \to 3\end{array}\right\}{\hspace{0.17em}}^9{C}_6\times 8=84\times 8=672$