For $A = \begin{{pmatrix}} a & b \\ c & d \end{{pmatrix}}$, eigenvalues solve $\lambda^2 - (\mathrm{{tr}} A)\lambda + \det A = 0$. Real or complex conjugate pairs are reported.
Eigenvalues of a 2×2 matrix
Matrix entries
What it computes. Roots of the characteristic polynomial for a single $2\times 2$ real matrix.
Limitations. Does not return eigenvectors (yet). Ill-conditioned cases can blur repeated roots. See the disclaimer.
Related: Eigenvalues in plain language · Eigenvalue · Linear algebra · 2×2 matrix toolkit