2×2 Matrix Calculator (Determinant & Inverse)
Compute determinant, inverse, trace, and eigenvalues of a 2×2 matrix.
What this calculates
The 2×2 matrix is the workhorse of linear algebra intuition: rotations, reflections, shears, and scalings all live here. This calculator computes determinant, inverse, trace, and eigenvalues for any 2×2 real matrix.
Formula & how it works
For M = [[a,b],[c,d]]: det = ad − bc. Inverse = (1/det) × [[d,−b],[−c,a]] if det ≠ 0. Trace = a + d. Eigenvalues = (trace ± √(trace² − 4det)) ÷ 2.
Worked example
M = [[3,2],[1,4]]. det = 12 − 2 = 10. Inverse = [[0.4,−0.2],[−0.1,0.3]]. Trace = 7. Eigenvalues = (7 ± √9)/2 = 5 and 2.
Frequently asked questions
What if det = 0?
Matrix is singular — no inverse exists, and columns are linearly dependent.
Complex eigenvalues?
Happen when trace² < 4·det. Indicates rotation in the transformation.
Bigger matrices?
Use the unit converter and statistics tools for related linear algebra. Larger matrices need different tools — this one stays 2×2 for clarity.