Matrix Trace Calculator

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The trace is only defined for square matrices (same number of rows and columns).
Row 1, Column 1 (diagonal)
Row 1, Column 2
Row 1, Column 3
Row 2, Column 1
Row 2, Column 2 (diagonal)
Row 2, Column 3
Row 3, Column 1
Row 3, Column 2
Row 3, Column 3 (diagonal)
Trace tr(A)
3

Sum of all main-diagonal elements a₁₁ + a₂₂ + ... + aₙₙ

Diagonal elementsa11 = 4, a22 = -3, a33 = 2
Matrix dimension3 x 3
Average diagonal element1
Trace tr(A)3
Average diagonal1

tr(A) = 3

  • The trace equals 3, which is the sum of all eigenvalues of this 3 x 3 matrix.
  • The average diagonal element is 1, which equals the average of all eigenvalues.
  • The trace is invariant under similarity transformations: tr(P⁻¹AP) = tr(A) for any invertible P.

Next stepA positive trace indicates the eigenvalues average to a positive value, but individual eigenvalues can still be negative.

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