Characteristic Polynomial Calculator

The characteristic polynomial of a square matrix A of size n x n is defined by the equation p(lambda) = det(A - lambda * I), where lambda represents the scalar variable and I is the identity matrix of the same dimension. For a 2x2 matrix, this expands into a quadratic equation based on the trace and determinant of the matrix: lambda^2 - tr(A)lambda + det(A) = 0. For larger 3x3 and 4x4 matrices, the calculation requires evaluating the determinant via cofactor expansion or row reduction, yielding higher-degree polynomial expressions whose roots correspond directly to the eigenvalues of the original matrix.

Consider a 2x2 matrix A with rows [2, 1] and [1, 2]. To find its characteristic polynomial, we set up the expression det(A - lambda * I). First, subtract lambda from the diagonal elements, resulting in the matrix with rows [2 - lambda, 1] and [1, 2 - lambda]. Next, calculate the determinant by multiplying the main diagonal entries and subtracting the product of the off-diagonal entries: (2 - lambda)(2 - lambda) - (1 * 1). Expanding this expression yields (4 - 4lambda + lambda^2) - 1, which simplifies to the final characteristic polynomial: lambda^2 - 4lambda + 3. Setting this polynomial to zero allows us to easily factor it into (lambda - 1)(lambda - 3) = 0, revealing the eigenvalues 1 and 3.

Always verify the dimensions of your input matrix before initiating the calculation, as characteristic polynomials only apply to square matrices where the number of rows equals the number of columns. Double-check your sign conventions when subtracting the scalar variable lambda along the principal diagonal to prevent downstream algebraic errors. When working with larger 4x4 matrices, consider utilizing sparse matrix properties or row reduction techniques if you are solving the equations by hand alongside this tool.