Question

Let \(M\) be any \(3\times3\) matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements of \(M^T M\) is seven, is __________.

Hint:

The Trace of $M^T M$ is always the sum of squares of all elements of $M$. Use multinomial coefficients to count permutations of the chosen entries.

Answer 1

Correct Answer: 540

Step 1: \(Tr(M^T M) = \sum_{i=1}^3 \sum_{j=1}^3 m_{ij}^2 = 7\).
Step 2: Let the 9 entries be \(a_1, a_2, ......., a_9 \in \{0, 1, 2\}\). We need \(\sum a_i^2 = 7\).
Step 3: Case 1: One entry is 2, three entries are 1, five entries are 0. Ways: \(\frac{9!}{1!3!5!} = \frac{9 \times 8 \times 7 \times 6}{6} = 504\).
Step 4: Case 2: Seven entries are 1, two entries are 0. Ways: \(\frac{9!}{7!2!} = \frac{9 \times 8}{2} = 36\).
Step 5: Total = \(504 + 36 = 540\).