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

Answer: (B)

$198$

Let $M$ be a $3\times 3$ square matrix,

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

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

The sum of the diagonal elements of $M^{T}M$ will be $S=a^2+b^2+c^2+d^2+e^2+f^2+g^2+h^2+i^2$

For $S=5\Rightarrow a^2+b^2+c^2+d^2+e^2+f^2+g^2+h^2+i^2=5$ out of $\{0,1,2\}$, there are only two possibilities:

1. One element is $2$, one element is $1$ and the rest are $0$ $={}^9{C}_{2}2!=72$

2. $5$ elements are $1$ and the rest are all $0$ $={}^9{C}_5=126$

Hence, total number of matrices $M$ $=72+126=198$

Hence, option $B$ is correct.