Matrix size
2x2
A=
 ⌈ a₁ b₁ ⌉ ⌊ a₂ b₂ ⌋
First row
a₁
b₁
Second row
a₂
b₂

# Characteristic Polynomial Calculator

By Anna Szczepanek, PhD candidate

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.

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

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

 ⌈ 2 3 ⌉ ⌊ 4 3 ⌋

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

 ⌈ 2 - λ 3 ⌉ ⌊ 4 3 - λ ⌋

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

 ⌈ 0 2 1 ⌉ | 1 3 -1 | ⌊ 2 0 2 ⌋

We need to calculate the determinant of

 ⌈ - λ 2 1 ⌉ | 1 3 - λ -1 | ⌊ 2 0 2 - λ ⌋

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:

`-λ³ + 5λ² - 2λ - 14`

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

 ⌈ a₁ b₁ c₁ ⌉ | a₂ b₂ c₂ ｜ ⌊ a₃ b₃ c₃ ⌋

`-λ³ + (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)ⁿλⁿ + (-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.

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

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

`det(A - λI) = det(Aᵀ - λI)`

2. 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 and the left one is not:

 ⌈ 2 1 ⌉ ⌊ 0 2 ⌋
 ⌈ 2 0 ⌉ ⌊ 0 2 ⌋
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

 ⌈ 2 3 ⌉ ⌊ 4 3 ⌋

It follows that `A²` is equal to

 ⌈ 16 15 ⌉ ⌊ 20 21 ⌋

and `5A` is equal to

 ⌈ 10 15 ⌉ ⌊ 20 15 ⌋

hence, and `A² - 5A` is equal to

 ⌈ 6 0 ⌉ ⌊ 0 6 ⌋

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

Anna Szczepanek, PhD candidate