Matrix Calculator
Enter a 2×2 matrix and get its determinant, trace, transpose, inverse and real eigenvalues with the full ad − bc working shown. Singular matrices (determinant 0) are detected and explained instead of crashing.
TheCalculators — Matrix Calculator
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Determinant
-14
det = ad − bc = 3·6 − 8·4 = 18 − 32
- Matrix
- [3, 8; 4, 6]
- Determinant (ad − bc)
- -14
- Trace (a + d)
- 9
- Transpose
- [3, 4; 8, 6]
- Invertible?
- Yes
- Inverse
- [-0.4286, 0.5714; 0.2857, -0.2143]
- Eigenvalues
- 10.3523 and -1.3523
Step-by-step
- 1Determinantdet = a·d − b·c = (3)·(6) − (8)·(4) = 18 − 32 = -14
- 2Tracetr = a + d = 3 + 6 = 9
- 3Adjugate (swap a↔d, negate b,c)adj = [6, -8; -4, 3]
- 4Inverse = adj ÷ detA⁻¹ = (1/-14) × adj = [-0.4286, 0.5714; 0.2857, -0.2143]
- 5Eigenvalues (characteristic eq.)λ² − 9λ + -14 = 0 → λ = (tr ± √(tr² − 4det)) ÷ 2 = 10.3523, -1.3523
Inverse matrix
Verify: A × A⁻¹ should give the identity matrix (1s on the diagonal).
Most useful next
Formula
det = ad − bc · A⁻¹ = 1/det × [d, −b; −c, a] · tr = a + d
Pro tips
- • A non-zero determinant means the matrix is invertible and maps the plane without collapsing it.
- • The determinant is also the signed area scale factor: |det| tells you how areas stretch.
- • Eigenvalues solve λ² − (trace)λ + det = 0 — for 2×2 matrices that is all you need.
FAQ
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