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Q.
The number of $3 \times 3$ matrices $M$ with entries from $\{0,1,2\}$ for which the sum of the diagonal entries of $M^TM$ is $5$, are
Matrices
Solution:
$\begin{bmatrix}a&b&c\\ d&e&f\\ g&h&i\end{bmatrix}\begin{bmatrix}a&d&g\\ b&e&h\\ c&f&i\end{bmatrix}$
Sum of diagonal elements,
$a^2 + b^2 + c^2 + d^2 + e^2 +f^2 + g^2 + h^2 + i^2 = 5$ Case - I Five (1's) and four (0's)
$\,{}^9C_5 = 126$ Case - II : One (2) and one (1)
$\,{}^9C_2 = 2! = 72$
$\therefore $ Total $=198$