Mathematics - Matrices Question with Solution | TestHub

Let,  $M=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]$

$\therefore$ Sum of the diagonal elements, $\mathrm{Tr}\left(M^T\text{}M\right)=a^2+b^2+c^2+d^2+e^2+f^2+g^2+h^2+i^2=5,$ where entries are $\left\{0,1,2\right\}$

Only two cases are possible.

(I) Five entries are $1$ and other fourare $0$

$\therefore { }^9{C}_5\times 1$

(II) One entry is $2$, one entry is $1$ and others are $0$.

$\therefore { }^9{C}_2\times 2$

Total cases$=126+72=198.$