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. In particular, there is a detailed explanation of how to find the characteristic polynomial of a 3x3 matrix.

If you're working with matrices, you might also find our eigenvalue and eigenvector 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 is the identity matrix of the size n × n (the same size as A); and
  • det is 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 that we 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 polynomial to the other by changing all their coefficients to opposite numbers (i.e., changing all of their signs). However, as long as we are concerned with the roots (i.e., the eigenvalues of A), it doesn't matter which definition you use.

As you now know what a characteristic polynomial is, why not look at some examples of how to find the characteristic polynomial?

How to use this characteristic polynomial calculator?

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

  1. Tell us the size of the matrix for which you want to find the characteristic polynomial.
  2. Enter all the coefficients of your matrix - row by row.
  3. Our characteristic polynomial calculator works as fast as lightning - the characteristic polynomial of your matrix appears at the bottom! ⚡
  4. Should you want to use the alternative definition of a characteristic polynomial, go to the advanced mode.

Characteristic polynomial of a 2×2 matrix

The determinant of

[aλbcdλ]\begin{bmatrix} a - \lambda & b \\ c& d - \lambda \end{bmatrix}

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 need to multiply the elements from its upper-left and bottom-right corners and subtract the product of the coefficients from the upper-right and bottom-left corners.

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 take a look at an example. We will find the characteristic polynomial of the following matrix:

[2343]\begin{bmatrix} 2 & 3 \\ 4& 3 \end{bmatrix}

As we explained in the first section, we need to calculate the determinant of

[2λ343λ]\begin{bmatrix} 2 - \lambda & 3 \\ 4& 3 - \lambda \end{bmatrix}

We have

(2-λ)(3-λ) - 3 * 4 = λ2 - 5λ - 6

Alternatively, we might have computed tr(A) = 2 + 3 = 5 and det(A) = 2 * 3 - 3 * 4 = -6.

Characteristic polynomial of a 3x3 matrix

Before we give you the general formula, let's solve an example.
We will find the characteristic polynomial of

[021131202]\begin{bmatrix} 0& 2& 1 \\ 1& 3& -1 \\ 2& 0& 2 \end{bmatrix}

We need to calculate the determinant of

[λ2113λ1202λ]\begin{bmatrix} - \lambda& 2& 1 \\ 1& 3- \lambda& -1 \\ 2& 0& 2- \lambda \end{bmatrix}

With the help of the Rule of Sarrus, we obtain:

-λ(3 - λ)(2 - λ) + 1×0×1 + 2×2×(-1) - 1×(3 - λ)×2 - (-1)×0×(-λ) - (2 - λ)×2×1

which simplifies to:

3 + 5λ2 - 2λ - 14

In general, the characteristic polynomial of a 3x3 matrix:

[a1b1c1a2b2c2a3b3c3]\begin{bmatrix} a_1 &b_1& c_1 \\ a_2 &b_2& c_2 \\ a_3 &b_3& c_3 \end{bmatrix}

reads:

3 + (a1 + b2 + c3)λ² - (a1b2 - a2b1 + a1c3 - a3c1 + b2c3 - b3c2)λ + (a1b2c3 - a2b3c1 + a3b1c2 - a3b2c1 - a1b3c2 - a2b1c3)

We see that the coefficient of λ² is just tr(A), and the intercept equals det(A). Moreover, the coefficient of λ is the sum of all of the 2x2 principal minors of A. This is a part of the general rule, as we will explain in the next section.

How to find the characteristic polynomial of larger matrices?

In general, the characteristic polynomial of an n x n matrix A has the form:

(-1)n×λn + (-1)n-1×S1×λn-1 + ... + (-1)k×Sn-k×λk + ... + Sn

where,

  • Sk is the sum of all k x k principal minors of A for k = 0, ..., n. In particular, S1 = tr(A) and Sn = det(A).

As you can see, it can be quite tedious to compute the characteristic polynomial by hand. That's why we've created this characteristic polynomial calculator! 😎

Properties of the characteristic polynomial of a matrix

Here are some useful properties of the characteristic polynomial of a matrix:

  1. A matrix is invertible (and so has full rank) if and only if its characteristic polynomial has a non-zero intercept. To find the inverse, you can use Omni's inverse matrix calculator.

  2. The degree of an eigenvalue of a matrix as a root of the characteristic polynomial is called the algebraic multiplicity of this eigenvalue.

  3. The matrix, A, and its transpose, Aᵀ, have the same characteristic polynomial:

    det(A - λI) = det(AT - λI)

  4. If two matrices are similar, then they have the same characteristic polynomial. However, the opposite is not true: two matrices with the same characteristic polynomial need not be similar!

    Example: The following two matrices both have a characteristic polynomial equal to (λ - 2)², but they are not similar since the right one is diagonalizable (well, in fact it is diagonal) and the left one is not:

[2102][2002]\quad \begin{bmatrix} 2& 1 \\ 0& 2 \end{bmatrix} \quad \begin{bmatrix} 2& 0 \\ 0& 2 \end{bmatrix}
  1. The Cayley–Hamilton theorem says that every matrix satisfies its own characteristic equation. More precisely: by replacing λ by A in the characteristic polynomial, we obtain the zero matrix (the intercept gets multiplied by the identity matrix).

Example:

We know that λ² - 5λ - 6 is the characteristic polynomial of

[2343]\quad \begin{bmatrix} 2& 3 \\ 4& 3 \end{bmatrix}

It follows that is equal to

[16152021]\quad \begin{bmatrix} 16& 15 \\ 20& 21 \end{bmatrix}

and 5A is equal to

[10152015]\quad \begin{bmatrix} 10& 15 \\ 20& 15 \end{bmatrix}

hence, A² - 5A is equal to

[6006]\quad \begin{bmatrix} 6& 0 \\ 0& 6 \end{bmatrix}

i.e., to 6I, and so A² - 5A - 6I is indeed the zero matrix.

Anna Szczepanek, PhD
Matrix size
2x2
A=
a₁b₁
a₂b₂
First row
a₁
b₁
Second row
a₂
b₂
Check out 34 similar linear algebra calculators 🔢
Adjoint matrixCholesky decompositionCofactor expansion… 31 more
People also viewed…

Black hole collision

The Black Hole Collision Calculator lets you see the effects of a black hole collision, as well as revealing some of the mysteries of black holes, come on in and enjoy!

Dot product

Dot product calculator finds the scalar product of two vectors, each one with three components.

Linear independence

The linear independence calculator is here to check whether your vectors are linearly independent and tell you the dimension of the space they span.

Lost socks

Socks Loss Index estimates the chance of losing a sock in the laundry.
Copyright by Omni Calculator sp. z o.o.
Privacy policy & cookies
main background