Question

How many $3\times 3$ matrices $M$ with entries from $\{0,1,2\}$ are there, for which the sum of the diagonal entries of $M^{T}M$ is $5?$

• A. $126$
• B. $198$
• C. $162$
• D. $135$

Answer 1

The correct option is B $198$

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

Sum of diagonal entries of $M^{T}M$
$=a^2+b^2+c^2+d^2+e^2+f^2+g^2+h^2+i^2=5$

Two possible case:
(1) Five "$1$'s" and four "$0$'s"
Number of ways $={}^9{C}_5=126$

(2) One "$1$" and one "$2$" and seven "$0$'s"
Number of ways$={}^9{C}_7\times {}^2{C}_1\times {}^1{C}_1=72$
Total number of possible matrices $=126+72=198$