The number of all $3 imes 3$ matrices $A$, with entries from the set $\{-1,0,1\}$ such that the su

Question

The number of all $3 	imes 3$ matrices $A$, with entries from the set $\{-1,0,1\}$ such that the sum of the diagonal elements of $A A^{T}$ is 3 , is

Professor PC Thomas Classes · Answer

(672) : Let $A=\left[a_{i j}ight]_{3 	imes 3}$ Now, sum of the diagonal elements of $A A^{T}$ is 3. i.e., $\operatorname{tr}\left(A A^{T}ight)=3$ [Given]\ $\Rightarrow a_{11}^{2}+a_{12}^{2}+a_{13}^{2}+a_{21}^{2}+\ldots+a_{33}^{2}=3$\ Possible cases are given as follows.\ $\left.\begin{array}{cc}0,0,0,0,0,0,1,1,1 & ightarrow 1 	ext { way } \ 0,0,0,0,0,0,-1,-1,-1 & ightarrow 1 	ext { way } \ 0,0,0,0,0,0,1,1,-1 & ightarrow 3 	ext { ways } \ 0,0,0,0,0,0,-1,1,-1 & ightarrow 3 	ext { ways }\end{array}ight\}$\ $	herefore$ Total number of ways\ $={ }^{9} C_{6} 	imes 8=84 	imes 8=672$\