The number of all 3 × 3 matrices A, with enteries from the set {–1, 0, 1} such that the sum of the diagonal elements of AA T is 3, is

Solution

Maths sir · 4 steps

The trace of AAᵀ is the sum of the squares of all nine entries, so just count the nonzero ones.

1. Diagonal of AAᵀ. The i-th diagonal entry of AAᵀ is row i of A dotted with itself, which is aᵢ₁² + aᵢ₂² + aᵢ₃². Adding all three diagonal entries gives the sum of the squares of every entry of A.

   tr(AAᵀ) = Σᵢ₌₁³Σⱼ₌₁³ aᵢⱼ² = 3

2. What that forces. Each entry is -1, 0 or 1, so each aᵢⱼ² is either 0 or 1. A total of 3 means exactly 3 of the 9 entries are nonzero and the other 6 are 0.
3. Count. Choose the 3 positions in C(9, 3) = 84 ways. Each of those 3 entries can be 1 or -1, giving 2³ = 8 sign choices.
4. So. The number of matrices is 84 × 8 = 672.

The common slip: 84. Choosing the 3 positions but forgetting that each nonzero entry can be 1 or -1 gives 84, not 672.

The idea to keep. Whenever you see the trace of AAᵀ or AᵀA, rewrite it as the sum of squares of all entries. That turns a matrix question into a counting question.

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