The total number of 3 x 3 matrices A having enteries
Question:

The total number of $3 \times 3$ matrices A having enteries from the set $(0,1,2,3)$ such that the sum of all the diagonal entries of $\mathrm{AA}^{\mathrm{T}}$ is 9 , is equal to__________.

Solution:

$\operatorname{Let} A=\left[\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right]$

diagonal elements of

$\mathrm{AA}^{\mathrm{T}}, \quad \mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}, \quad \mathrm{~d}^{2}+\mathrm{e}^{2}+\mathrm{f}^{2}, g^{2}+b^{2}+c^{2}$

Sum $=a^{2}+b^{2}+c^{2}+d^{2}+e^{2}+f^{2}+g^{2}+h^{2}+i^{2}=9$

$\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}, \mathrm{i} \in\{0,1,2,3\}$