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Trace of a 2×2 Matrix

tr⁡(M)=a11+a22\operatorname{tr}(M) = a_{11} + a_{22}

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The trace is the sum of the diagonal, and nothing could look more arbitrary — why should those particular entries, which depend entirely on the coordinate axes you happened to pick, add up to anything meaningful? Yet they do. Change basis and every entry of the matrix changes, but the trace does not move, because it equals the sum of the eigenvalues. Alongside the determinant, which equals their product, the trace pins down a 2×2 matrix's characteristic equation completely: λ² − tr(M)λ + det(M) = 0.

That pairing is the working tool. For rows (3, 1) and (2, −7) the trace is 3 + (−7) = −4; with the determinant at −23, the eigenvalues are the roots of λ² + 4λ − 23 = 0 — no elimination required. Physicists meet the same invariance as the trace of the stress tensor, which is three times the mean pressure whichever way the axes are turned. James Joseph Sylvester coined the word matrix for these arrays in 1850, meaning a womb from which determinants are born; the trace was one of the first invariants he and Cayley catalogued.

Trace of a 2×2 Matrix
tr⁡(M)=a11+a22\operatorname{tr}(M) = a_{11} + a_{22}
Where
  • tr⁡(M)\operatorname{tr}(M)= Trace
  • a11a_{11}= Top-left entry
  • a22a_{22}= Bottom-right entry
Missing one of these? Work it out first, then come back