Math Problem Statement
Verify cayley Hamilton theorem and find A inverse
Verify Cayley-Hamilton theorem and find A inverse.
Solution
The matrix you provided is:
0 & 0 & 1 \\ 1 & 2 & 0 \\ 2 & -1 & 0 \end{pmatrix}$$ ### Cayley-Hamilton Theorem The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation. #### Step 1: Find the Characteristic Polynomial The characteristic polynomial $$p(\lambda)$$ of matrix $$A$$ is given by: $$p(\lambda) = \det(A - \lambda I)$$ Where $$I$$ is the identity matrix and $$\lambda$$ is a scalar. #### Step 2: Verify Cayley-Hamilton Theorem Once we find the characteristic polynomial, we can substitute $$A$$ into it and verify that $$p(A) = 0$$. #### Step 3: Find the Inverse of $$A$$ The inverse of a matrix $$A^{-1}$$ can be found using various methods such as the adjugate formula or row reduction. I'll proceed step by step. I'll calculate both the characteristic polynomial and the inverse now.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrices
Matrix Inversion
Formulas
Characteristic polynomial: p(λ) = det(A - λI)
Matrix Inverse formula using adjugate: A^(-1) = (1/det(A)) * adj(A)
Theorems
Cayley-Hamilton Theorem
Suitable Grade Level
Undergraduate level (Linear Algebra)
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