Eigenvalues of a 2×2 Matrix - Linear Algebra Step by Step

Solve the characteristic polynomial λ² - tr(A)λ + det(A) = 0 by factoring. Verify that each eigenvalue satisfies p(λ) = 0.

Example

Solve the characteristic quadratic. The eigenvalues of a matrix are the numbers for which the matrix, minus that value times the identity, becomes singular, and setting that determinant to zero gives the characteristic equation. For a two-by-two matrix this is a quadratic whose roots are the eigenvalues, the special scaling factors along the matrix's own directions.

highlighted = computed this step

Step 1 — Set up

Set up the given matrix data.

A=\begin{bmatrix}4&1\\2&3\end{bmatrix}

Step 2 — Trace and determinant

Record trace and determinant: 7,10.

tr_det=\hl{(7,10)}

Step 3 — Solve quadratic

Solve the quadratic roots: 2,5.

roots=\hl{(2,5)}

Step 4 — Eigenvalues

The eigenvalues are 2 and 5.

\lambda=\hl{2,5}

eigenvalue A scalar λ is an eigenvalue of A if Av = λv for some nonzero vector v. Eigenvalues are roots of the characteristic polynomial det(A - λI) = 0.