The number of 3 × 3 matrices M with entries from 0,1, 2 , such that...

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

$∴$ Sum of the diagonal elements, $Tr ( {M}^{T}\text{}M ) = {a}^{2} + {b}^{2} + {c}^{2} + {d}^{2} + {e}^{2} + {f}^{2} + {g}^{2} + {h}^{2} + {i}^{2} = 5,$ where entries are \left\{0, 1, 2\right\}$</p><p>Only two cases are possible.</p><p>(I) Five entries are$1 and other fourare $0$

$∴ {}^{9}{C}_{5} \times 1$

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

$∴ {}^{9}{C}_{2} \times 2$

Total cases$= 126 + 72 = 198 .$