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

Answer 2

Let matrix $M =\left[ k _{ i } j \right]$
Then sum of diagonal entries $=\sum k _{ ij }^{2}$
$\Rightarrow \sum k _{ ij }^{2}=5$
where $k _{ ij }$ are from $\\{0,1,2\\}$
$\Rightarrow$ Total number of matrices $={ }^{9} C _{5}+{ }^{9} C _{1} \cdot{ }^{8} C _{1}=198$