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Matrix Trace Calculator

Enter a 2×2 matrix to compute its trace and determinant.

How it is calculated

The matrix is entered as four numbers in the layout [[a, b], [c, d]]: a and b are the top row, c and d the bottom row. The trace is the sum of the main diagonal, tr = a + d, and the determinant is det = a × d − b × c. Negative numbers and decimals are accepted, and all four boxes must be filled. Results are shown without trailing zeros, up to four decimals. Calculations use ordinary double-precision numbers, so values beyond about 15 significant digits are approximate, and numbers so large that the trace or determinant overflows give no result. The tool covers 2×2 matrices only. The trace and determinant also give the eigenvalues λ as roots of λ² − tr·λ + det = 0, but this tool does not solve that equation, so you do it yourself. A determinant of 0 means the matrix is singular.

Worked example

Take the matrix with a = 3, b = 1, c = 2, d = 4. Trace = 3 + 4 = 7. Determinant = 3 × 4 − 1 × 2 = 12 − 2 = 10. If you want the eigenvalues, solve λ² − 7λ + 10 = 0, which factorises as (λ − 2)(λ − 5) = 0, so the eigenvalues are 2 and 5. They add up to 7, the trace, and multiply to 10, the determinant.

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Common mistakes

  • Adding the off-diagonal entries. The trace uses only a and d, the top-left and bottom-right numbers, never b and c.
  • Putting the entries in column order. The boxes follow the rows, so the second box is the top-right number, not the bottom-left.
  • Computing the determinant as a × d + b × c. The off-diagonal product is subtracted, not added.
  • Expecting a 3×3 matrix to work. The tool handles 2×2 only, and a larger matrix needs a different calculation.
  • Leaving one of the four boxes empty and assuming it counts as 0. An empty entry gives no result, so type 0 explicitly.

About this calculator

The trace of a square matrix is simply the sum of the entries on its main diagonal. For a 2×2 matrix with entries a, b, c, d arranged as [[a, b], [c, d]], the trace is a + d, while the determinant, ad − bc, measures how the matrix scales area.

Together, trace and determinant summarise a lot about a matrix's behaviour without needing to fully diagonalise it: for a 2×2 matrix, the trace equals the sum of its two eigenvalues and the determinant equals their product, so these two numbers alone are enough to find the eigenvalues by solving a simple quadratic equation. This makes trace and determinant everyday tools in linear algebra courses, in physics and engineering for analysing linear transformations and stability of systems, and in computer graphics for understanding what a transformation matrix does to shapes.

Enter a 2×2 matrix and this calculator computes both its trace and its determinant.

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