Linear Algebra Intermediate

Matrix Trace Calculator

Compute tr(A) = sum of diagonal entries. For 2×2 matrices, verify tr(A) = λ₁ + λ₂ via the characteristic polynomial. Enter a second matrix B to demonstrate tr(A+B) = tr(A) + tr(B).

Subject: Linear Algebra
Input: 2×2 or 3×3 matrices A and B
Output: tr(A), eigenvalue sum, properties
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Matrix Input
Result
Enter matrix values and click Calculate

Step-by-Step Work

What the Trace Means
tr(A) = Σ aᵢᵢ = λ₁ + λ₂ + ⋯ + λₙ

The trace is the sum of diagonal entries and equals the sum of all eigenvalues (counted with multiplicity). It appears in the characteristic polynomial as the coefficient of λⁿ⁻¹.

The trace is preserved under similarity transformations: tr(P⁻¹AP) = tr(A).

Exam tip — for a 2×2 matrix, the characteristic polynomial is λ² − tr(A)·λ + det(A) = 0.
Trace Properties
  • tr(A + B) = tr(A) + tr(B)
  • tr(kA) = k · tr(A)
  • tr(Aáµ€) = tr(A)
  • tr(AB) = tr(BA) (cyclic)
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