Free Characteristic Polynomial Calculator

The characteristic polynomial is a fundamental construct in linear algebra, linking a square matrix to its eigenvalues. With the Characteristic Polynomial Calculator (a free online matrix polynomial solver), you can instantly find the characteristic polynomial for 2×2, 3×3, and 4×4 matrices, saving the tedious manual expansion. In this article, we will revisit the definition, provide step‑by‑step formulas for various sizes, and discuss key properties — all while demonstrating how the solver makes the process effortless.

Definition of the Characteristic Polynomial

For an $n \times n$ matrix $A$, the characteristic polynomial in variable $\lambda$ is defined as:

$$p(\lambda) = \det(A - \lambda I)$$

where $I$ is the identity matrix of size $n$ and $\det$ denotes the determinant. Because the roots of $p(\lambda)$ are precisely the eigenvalues of $A$, this polynomial is also referred to as the eigenvalue polynomial. Some textbooks adopt $p(\lambda) = \det(\lambda I - A)$ instead; this alternative version equals $(-1)^n \det(A - \lambda I)$. Consequently, the two polynomials are identical when $n$ is even, and have opposite signs for all coefficients when $n$ is odd. Importantly, the eigenvalues remain unaffected by the sign convention.

How to Find the Characteristic Polynomial with the Solver

Using this calculator is straightforward:

1. Choose the matrix size: 2×2, 3×3, or 4×4.
2. Enter the entries of the matrix row by row into the input fields.
3. The calculator displays the polynomial immediately.
4. Optionally, toggle between the $A - \lambda I$ and $\lambda I - A$ conventions by changing the "definition variant" setting.

This eliminates the need to memorize complex formulas, especially for larger matrices.

Characteristic Polynomial of a 2×2 Matrix

For a general $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the characteristic polynomial expands to:

$$(a - \lambda)(d - \lambda) - bc = \lambda^2 - (a + d)\lambda + (ad - bc).$$

It can be written compactly as $\lambda^2 - \operatorname{tr}(A)\lambda + \det(A)$, where $\operatorname{tr}(A) = a + d$ and $\det(A) = ad - bc$.

Example: Take $A = \begin{pmatrix} 2 & 3 \\ 4 & 3 \end{pmatrix}$. Then $\operatorname{tr}(A) = 5$ and $\det(A) = 2\cdot 3 - 3\cdot 4 = -6$. Hence the characteristic polynomial is $\lambda^2 - 5\lambda - 6$. Direct expansion using $(2-\lambda)(3-\lambda) - 3\cdot 4$ gives the same result, confirming the formula.

Characteristic Polynomial of a 3×3 Matrix

For a $3\times 3$ matrix, expansion can be performed using the Rule of Sarrus or cofactor expansions. Consider the matrix:

$$A = \begin{pmatrix} 0 & 1 & 2 \\ 2 & 3 & -1 \\ 1 & 0 & 2 \end{pmatrix}.$$

Compute $\det(A - \lambda I)$:

$$\det\begin{pmatrix} -\lambda & 1 & 2 \\ 2 & 3-\lambda & -1 \\ 1 & 0 & 2-\lambda \end{pmatrix}.$$

Applying the Rule of Sarrus yields:

$$-\lambda(3-\lambda)(2-\lambda) + 1\cdot0\cdot1 + 2\cdot2\cdot(-1) - 1\cdot(3-\lambda)\cdot2 - (-1)\cdot0\cdot(-\lambda) - (2-\lambda)\cdot2\cdot1,$$

which simplifies to:

$$-\lambda^3 + 5\lambda^2 - 2\lambda - 14.$$

In general, for a 3×3 matrix with entries $a_{ij}$, the characteristic polynomial can be expressed as:

$$-\lambda^3 + (a_{11}+a_{22}+a_{33})\lambda^2 - \bigl( \text{sum of all principal } 2\times 2 \text{ minors} \bigr)\lambda + \det(A).$$

The coefficient of $\lambda^2$ is the trace, and the constant term is the determinant.

General Formula for $n \times n$ Matrices

For an $n \times n$ matrix, the characteristic polynomial takes the form:

$$p(\lambda) = (-1)^n \lambda^n + (-1)^{n-1} S_1 \lambda^{n-1} + \cdots + (-1)^k S_{n-k} \lambda^k + \cdots + S_n,$$

where $S_k$ is the sum of all $k\times k$ principal minors of $A$. In particular, $S_1 = \operatorname{tr}(A)$ and $S_n = \det(A)$. For matrices larger than 3×3, calculating these sums by hand becomes quickly impractical, making a characteristic polynomial solver invaluable.

Important Properties of the Characteristic Polynomial

• Invertibility: A matrix is invertible if and only if the constant term of its characteristic polynomial (i.e., $S_n$) is non‑zero. This is equivalent to $\det(A) \neq 0$.
• Algebraic Multiplicity: The multiplicity of an eigenvalue as a root of $p(\lambda)$ is called its algebraic multiplicity.
• Transpose: $A$ and its transpose $A^\mathsf{T}$ share the same characteristic polynomial.
• Similarity: Similar matrices have identical characteristic polynomials, but the converse is not true. For example, the matrices $\begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}$ and $\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$ both have characteristic polynomial $(\lambda-2)^2$, yet they are not similar because the first is not diagonalizable while the second is diagonal.
• Cayley‑Hamilton Theorem: Every square matrix satisfies its own characteristic equation. That is, if you substitute $A$ for $\lambda$ in $p(\lambda)$ (interpreting the constant term as $cI$), you obtain the zero matrix.

Illustration of Cayley‑Hamilton: For the earlier 2×2 matrix $A = \begin{pmatrix}2 & 3 \\ 4 & 3\end{pmatrix}$ with characteristic polynomial $\lambda^2 - 5\lambda - 6$, we have

$$A^2 = \begin{pmatrix}16 & 15 \\ 20 & 21\end{pmatrix},\quad 5A = \begin{pmatrix}10 & 15 \\ 20 & 15\end{pmatrix},\quad 6I = \begin{pmatrix}6 & 0 \\ 0 & 6\end{pmatrix}.$$

Therefore,

$$A^2 - 5A - 6I = \begin{pmatrix}16-10-6 & 15-15-0 \\ 20-20-0 & 21-15-6\end{pmatrix} = \begin{pmatrix}0 & 0 \\ 0 & 0\end{pmatrix},$$

confirming the theorem.