Let M be any 3 × 3 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 ________.

Solution

Maths sir · 3 steps

The trace of MᵀM is just the sum of the squares of all nine entries, so this is a counting problem.

1. Read the trace. The diagonal entry (MᵀM)ᵢᵢ is the sum of the squares of the entries in column i of M. Adding all three diagonal entries therefore adds the square of every entry of M.

   tr(MᵀM) = Σᵢ₌₁³Σⱼ₌₁³ mᵢⱼ² = 7

2. Possible squares. Each entry is 0, 1 or 2, so its square is 0, 1 or 4. You need nine such squares adding to 7. Two 4s already make 8, so at most one entry can be 2.

   Case	Entries used	Number of matrices
   No 2	seven 1s, two 0s	C(9, 7) = 36
   One 2	one 2, three 1s, five 0s	9 × C(8, 3) = 9 × 56 = 504

3. So. The total number of matrices is 36 + 504 = 540.

The common slip: 36. Forgetting that one entry can be 2 (since 4 + 1 + 1 + 1 = 7) leaves only the all-ones case, which gives 36, not 540.

The idea to keep. Whenever you see tr(MᵀM) or tr(MMᵀ), replace it with the sum of the squares of all the entries. Then just split into cases and count.

Answer from the NTA final answer key. Solution by PrepWiser.