Matrix Determinant Calculator – 2×2 with Extras

This determinant tool wraps the 2×2 core with the quantities that depend on it. For A = [[3,1],[2,4]]: det = 3×4 − 1×2 = 10, trace = 3+4 = 7, and the characteristic equation λ² − 7λ + 10 = 0 factors as (λ−5)(λ−2), giving eigenvalues 5 and 2 — the calculator solves this quadratic for every input. The determinant’s meaning prints alongside: |det| = 10 is the area scale the matrix applies, sign −1 would flag an orientation flip, and det = 0 declares the matrix singular with no inverse. Invertibility check and the inverse (1/10)[[4,−1],[−2,3]] appear when valid. The output pairs exact fractions with decimals, notes the eigen-sum equals trace and eigen-product equals determinant (5+2 = 7 ✓, 5×2 = 10 ✓), and flags matrices with complex eigenvalues when the discriminant trace² − 4det goes negative.

1. 1

   Enter the 2×2 entries

   a, b, c and d

2. 2

   Compute det and trace

   10 and 7 for the example, both shown

3. 3

   Solve the characteristic equation

   λ² − 7λ + 10 = 0 → λ = 5, 2

4. 4

   Interpret

   Area scale, orientation, invertibility and the inverse matrix

• 1

  Eigenvalues sum to the trace and multiply to the determinant

• 2

  det 0 = singular: no inverse exists

• 3

  Negative determinant means orientation reversal

• Computing bc − ad and flipping the sign

• Reading eigenvalues as entries of the matrix

• Assuming a negative determinant blocks inversion — only zero does

• Ignoring complex eigenvalues when trace² < 4det