Characteristic Polynomial Calculator — Find p(λ) = det(A − λI) of a 2×2, 3×3 or 4×4 Matrix

Characteristic Polynomial Calculator

With the help of our characteristic polynomial calculator, you can quickly determine the characteristic polynomial of a 2×2, 3×3, or 4×4 matrix. Do you want to learn how to find the characteristic polynomial? Keep reading, as we first recall what a characteristic polynomial is and then give the formulas as well as examples — including a detailed explanation of how to find the characteristic polynomial of a 3×3 matrix.

If you're working with matrices, you might also find our eigenvalue and eigenvector calculator, the matrix determinant calculator, and the diagonalize-matrix calculator useful.

What is a characteristic polynomial?

We define the characteristic polynomial, p(λ), of a square matrix A of size n × n as:

p(λ) := det(A − λI)

where:

• I — the identity matrix of size n × n (the same size as A); and
• det — the determinant of a matrix (see the matrix determinant calculator if you're not sure what we mean).

Keep in mind that some authors define the characteristic polynomial as det(λI − A). We can obtain that polynomial from the one defined above by multiplying it by (−1)ⁿ. Hence, those polynomials coincide if n is even. If n is odd, you can go from one to the other by changing all of its coefficients to the opposite sign. However, as long as we are concerned with the roots (i.e., the eigenvalues of A), it doesn't matter which definition you use. This calculator lets you pick either variant with the definition setting.

How to use this characteristic polynomial calculator

You don't need to be a math expert to use the characteristic polynomial calculator. Just follow the steps below:

• Tell us the size of the matrix for which you want to find the characteristic polynomial.
• Enter all the coefficients of your matrix — row by row.
• Our characteristic polynomial calculator works as fast as lightning — the characteristic polynomial of your matrix appears on the right! ⚡
• Should you want to use the alternative definition of a characteristic polynomial, change the definition variant setting.

Characteristic polynomial of a 2×2 matrix

The determinant of

| a − λ      b     |
| c          d − λ |

is equal to

(a − λ)(d − λ) − bc = λ² − (a + d)λ + (ad − bc)

This is because, to find the determinant of a 2×2 matrix, you multiply the elements on the main diagonal and subtract the product of the off-diagonal elements. Observe that we can write the characteristic polynomial of a 2×2 matrix A as:

λ² − tr(A)λ + det(A)

where tr(A) is the trace of A, i.e., the sum of the diagonal elements of A.

Example. Let us find the characteristic polynomial of the matrix

| 2  3 |
| 4  3 |

Here tr(A) = 2 + 3 = 5 and det(A) = 2·3 − 3·4 = −6, so the characteristic polynomial is λ² − 5λ − 6 = (λ − 6)(λ + 1). The eigenvalues are therefore λ = 6 and λ = −1.

Characteristic polynomial of a 3×3 matrix

For a 3×3 matrix A, the characteristic polynomial p(λ) = det(λI − A) works out to:

λ³ − tr(A)λ² + (M₁₁ + M₂₂ + M₃₃)λ − det(A)

where M_ii are the three 2×2 principal minors obtained by deleting the i-th row and the i-th column. The coefficient of λ² is minus the trace, the coefficient of λ is the sum of principal 2×2 minors, and the constant term is minus the determinant. Our calculator performs this computation exactly and lists every coefficient for you.

How to find the characteristic polynomial of larger matrices

For a general n × n matrix, the coefficients of the characteristic polynomial are the elementary symmetric functions of the eigenvalues, which equal the sums of the k×k principal minors of A (up to sign). Computing them directly by expanding det(λI − A) becomes tedious, so this calculator uses the Faddeev–LeVerrier algorithm — an elegant recurrence that produces every coefficient using only matrix multiplications and traces, and stays exact for integer matrices.

Properties of the characteristic polynomial of a matrix

• Its degree equals the size of the matrix, n, and (for the det(λI − A) form) it is monic — the leading coefficient is 1.
• The roots of the characteristic polynomial are exactly the eigenvalues of A.
• The sum of the eigenvalues equals the trace tr(A) (the coefficient of λⁿ⁻¹ is −tr(A)).
• The product of the eigenvalues equals the determinant det(A) (the constant term is (−1)ⁿdet(A)).
• Similar matrices (A and P⁻¹AP) share the same characteristic polynomial, so it is an invariant of the linear map.
• By the Cayley–Hamilton theorem, every matrix satisfies its own characteristic polynomial: p(A) = 0.

Enter your own numbers into the grid on the left, choose the size of your matrix (2×2, 3×3, or 4×4) and the definition variant, then press Find characteristic polynomial. You can work with pure abstract numbers, or attach a metric, US customary, or currency context (12 world currencies including the US dollar and the Russian ruble) when your matrix entries represent real-world quantities.