Enter a 2×2 matrix and instantly get both eigenvalues, with an eigenvector for each.
Eigenvalue & Eigenvector Calculator (2×2)
How It’s Calculated
Eigenvalues satisfy det(A − λI) = 0, which for a 2×2 matrix reduces to λ² − (trace)λ + (determinant) = 0 — the trace is a+d, the determinant is ad−bc. Solving that quadratic gives both eigenvalues. For each λ, the eigenvector is any nonzero solution to (A − λI)v = 0; this calculator picks the simplest such vector directly from the matrix entries.
Worked Example
For A = [[4,1],[2,3]]: trace = 7, det = 12−2 = 10. λ²−7λ+10=0 factors to (λ−5)(λ−2)=0, so λ = 5 and λ = 2. For λ=5: eigenvector (1, 1). For λ=2: eigenvector (1, −2).
Frequently Asked Questions
Q: What does a negative discriminant mean here?
The matrix has complex eigenvalues — geometrically, it represents a rotation (possibly combined with scaling) rather than a simple stretch along real directions.
Q: Is the eigenvector this tool gives the only correct one?
No — any nonzero scalar multiple of an eigenvector is also a valid eigenvector for the same eigenvalue. This tool shows one representative vector; your textbook’s answer may be a scaled version of the same direction.
Eigenvalues and eigenvectors are core to the Linear Algebra module in VTU Engineering Mathematics 1. See how Daniel Classes teaches VTU Engineering Mathematics →