Characteristic Polynomial - Linear Algebra Step by Step

Form the characteristic polynomial det(A - λI) = λ² - tr(A)λ + det(A) for a 2×2 matrix. Compute the trace and determinant, then assemble the polynomial.

Example

Use trace and determinant to form the polynomial.

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

Compute the trace: tr = 7.

tr=\hl{7}

Step 3 — Determinant

Compute the determinant: det = 10.

det=\hl{10}

Step 4 — Characteristic polynomial

Form the characteristic polynomial from trace 7 and determinant 10.

\chi_A(\lambda)=\lambda^2 - 7\lambda + 10

characteristic polynomial For an n×n matrix A, the characteristic polynomial is det(A - λI). For 2×2: det(A - λI) = λ² - tr(A)λ + det(A), where tr(A) = a₁₁ + a₂₂ and det(A) = a₁₁a₂₂ - a₁₂a₂₁.