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The characteristic polynomial calculator computes the characteristic polynomial of a square matrix of any order (2×2, 3×3, 4×4, etc.). It helps you quickly find the eigenvalues and simplifies solving determinant equations for control systems or linear algebra problems.
The characteristic polynomial of a square matrix is a polynomial whose roots are the eigenvalues of the matrix. It is defined as:
f(λ) = det(A − λI) = 0
Where:
The characteristic polynomial is obtained by calculating the determinant of (A − λI). Solving this polynomial gives the eigenvalues of the matrix.
Find the characteristic polynomial of the matrix:
\[ A = \begin{bmatrix} 3 & 5 & 2 \\ 7 & 9 & 3 \\ 4 & 5 & 4 \end{bmatrix} \]
Step 1: Subtract λ from the diagonal elements of the matrix:
\[ A - \lambda I = \begin{bmatrix} 3-\lambda & 5 & 2 \\ 7 & 9-\lambda & 3 \\ 4 & 5 & 4-\lambda \end{bmatrix} \]
Step 2: Compute the determinant of (A − λI) to get the characteristic polynomial:
\[ \text{Characteristic Polynomial} = -\lambda^3 + 16\lambda^2 - 17\lambda - 19 \]
This polynomial can then be solved to find the eigenvalues of the matrix.
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