Matrix Calculator

1. Definition & Formula: What is the trace of a matrix?

The trace of an $n\times n$ square matrix $A=[a_{ij}]$ is the sum of its diagonal entries:

Trace is only defined for square matrices and is a simple, useful invariant in linear algebra.

Key properties:

• Linearity: $\mathrm{tr}(A+B)=\mathrm{tr}(A)+\mathrm{tr}(B)$ and $\mathrm{tr}(cA)=c\,\mathrm{tr}(A)$.
• Similarity invariance: if $B=P^{-1}AP$, then $\mathrm{tr}(B)=\mathrm{tr}(A)$.
• Cyclic property: $\mathrm{tr}(AB)=\mathrm{tr}(BA)$ (when dimensions agree).

2. How to compute trace (quick rule)

To compute the trace, simply add the diagonal entries of a square matrix. You do not need row reduction, determinants, or inversion.

3. Worked examples

Example 1 (2×2)

Compute the trace:

Example 2 (3×3)

Compute the trace:

4. Common mistakes

• Applying trace to non-square matrices: trace is not defined for non-square matrices.
• Including off-diagonal entries: only add diagonal elements $a_{11},a_{22},\dots$.
• Forgetting signs: negative diagonal entries must be included correctly.
• Expecting trace to equal determinant: trace and determinant are different invariants.

5. Practice problems (answers hidden)

Try these; click Show Answer to reveal the trace.

Exercise 1

Exercise 2

Exercise 3

Exercise 4

The trace of a matrix is also connected with the matrix inverse, matrix rank, and eigenvectors calculator.