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For a 3 × 3 matrix A = (aᵢⱼ), the trace T₃(A) is defined as a₁₁ + a₂₂ + a₃₃, that is, the sum of the diagonal entries. Which of the following statements is not necessarily true?

1. T₃(kA) = kT₃(A), where k is a scalar
2. T₃(A + B) = T₃(A) + T₃(B)
3. T₃(I₃) = 3
4. T₃(A²) = [T₃(A)]²

Correct answer: T₃(A²) = [T₃(A)]²

Solution

The statement T₃(A²) = [T₃(A)]² is not necessarily true because the trace of the product of two matrices does not equal the product of their traces. In general, T₃(A²) equals T₃(A) when A is a diagonal matrix, but this does not hold for all matrices.

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