Characteristic Polynomial Calculator

Find the characteristic polynomial, trace, and determinant of a 2 by 2 matrix.

Calculate a Characteristic Polynomial

Enter the four entries of a 2 by 2 square matrix.

Row 1, Column 1

Row 1, Column 2

Row 2, Column 1

Row 2, Column 2

About the Characteristic Polynomial Calculator

The characteristic polynomial packages essential information about a square matrix into one polynomial. Its roots are the matrix's eigenvalues, which describe directions or states that are scaled without changing orientation under the associated linear transformation. This calculator evaluates the polynomial for a real 2 by 2 matrix and also reports its trace and determinant. For a matrix with entries a and b in the first row and c and d in the second row, the monic characteristic polynomial is p(λ) = det(λI - A). Expanding the determinant gives λ squared minus (a + d) times λ plus (ad - bc). The quantity a + d is the trace, while ad - bc is the determinant. For a 2 by 2 matrix, those two values completely determine the characteristic polynomial. Some textbooks define the polynomial as det(A - λI) instead. For even-sized 2 by 2 matrices this produces the same polynomial, so there is no sign ambiguity here. In odd dimensions the two conventions differ by an overall negative sign, but their roots remain identical. The monic convention used here keeps the leading coefficient equal to one and is common in linear algebra. Characteristic polynomials support eigenvalue calculations, stability analysis, differential equations, recurrence relations, quantum mechanics, control systems, and data science. A diagonal matrix provides the simplest example: its eigenvalues are the diagonal entries, and its polynomial is the product of λ minus each diagonal value. A repeated root can indicate a repeated eigenvalue, though further work is needed to determine whether enough independent eigenvectors exist. Enter ordinary integers or decimals in reading order across the matrix. The calculator computes the trace and determinant first, then places them into the general quadratic form. Decimal arithmetic may be rounded for display, so exact fractional work may still be preferable in symbolic proofs. To find numerical eigenvalues afterward, solve the displayed quadratic equation by factoring or applying the quadratic formula.

Characteristic Polynomial Examples

Trace supplies the linear coefficient and determinant supplies the constant term.

Matrix	Polynomial	Explanation
[2, 0; 0, 3]	λ² - 5λ + 6	The polynomial factors as (λ - 2)(λ - 3).
[1, 2; 3, 4]	λ² - 5λ - 2	The trace is 5 and the determinant is -2.
[0, -1; 1, 0]	λ² + 1	The trace is zero and determinant is one.

How to Calculate a Characteristic Polynomial

Enter the first matrix row from left to right.
Enter the second matrix row in the remaining fields.
Select Calculate Polynomial to compute trace and determinant.
Read the monic polynomial and solve it for eigenvalues if needed.

Characteristic Polynomial FAQ

What are the roots of a characteristic polynomial?

Its roots are the eigenvalues of the matrix. Each root identifies a scale factor associated with at least one eigenvector.

How do trace and determinant appear in the polynomial?

For a 2 by 2 matrix, the trace is the negative of the linear coefficient. The determinant is the constant term.

Does every square matrix have a characteristic polynomial?

Yes, every finite square matrix has one. An n by n matrix produces a polynomial of degree n.

Why must the matrix be square?

The expression subtracts a scalar identity matrix and takes a determinant, both requiring equal row and column counts. Rectangular matrices do not have characteristic polynomials in this standard sense.

Can a characteristic polynomial have complex roots?

Yes, a real matrix may have nonreal eigenvalues in conjugate pairs. For example, a planar rotation matrix can produce λ squared plus one.