Question

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

• A. 198
• B. 162
• C. 126
• D. 135

Answer 1

The correct option is A 198

Sum of diagonal entries of $M^{T}M$ is $\sum a_i^2$
${\sum }_{i=1}^9{a}_i^2=5$
Possibilities
I. 2, 1, 0, 0, 0, 0, 0, 0, 0, which gives $\frac{9!}{7!}$ matrices
II. 1, 1, 1, 1, 1, 0, 0, 0, 0, which gives $\frac{9!}{4!\times 5!}$ matrices Total matrices $=9\times 8+9\times 7\times 2=198$