For a square matrix A, the characteristic polynomial is built by subtracting a scalar multiple of the identity matrix and taking the determinant. In this calculator, the convention is det(?I - A), which is one of the standard ways to write it.

The roots of that polynomial are the eigenvalues of the matrix. If you have seen the alternative form det(A - ?I), that is the same information with the signs arranged differently.

The characteristic polynomial gives you a direct route from a matrix to its eigenvalues. That is useful in linear algebra, but it also shows up in stability questions, differential equations, and diagonalization problems.

The key idea is that the polynomial packages a lot of matrix behavior into one expression, and the zeros of that expression tell you which values make the matrix act in a special way.

For a 2x2 matrix, the characteristic polynomial has the form ?� - tr(A)? + det(A). That means the trace appears in the middle coefficient and the determinant appears as the constant term.

This is why the calculator shows those values alongside the polynomial. They make it easy to check whether the result has the right shape without expanding everything by hand.

For a 3x3 matrix, the pattern becomes ?� - tr(A)?� + S? - det(A), where S is the sum of the 2x2 principal minors. The middle coefficient is the part that usually takes the most work to compute by hand.

That is one reason the calculator is helpful: it handles the coefficient pattern for you and presents the polynomial in a form that is easy to compare with the matrix summary values.