Math Problem Statement
Q.9. Verify Cayley-Hamilton theorem for the matrix A = [[1, 1, 1], [1, 2, 1], [3, 2, 3]]
Solution
The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation. To verify the theorem for the given matrix , we need to follow these steps:
Given:
1 & 1 & 1 \\ 1 & 2 & 1 \\ 3 & 2 & 3 \end{bmatrix}$$ ### Step 1: Find the Characteristic Polynomial of $$A$$ The characteristic polynomial of matrix $$A$$ is given by: $$p(\lambda) = \det(A - \lambda I)$$ where $$I$$ is the identity matrix of the same order as $$A$$ (in this case, $$3 \times 3$$), and $$\lambda$$ is the eigenvalue variable. First, compute $$A - \lambda I$$: $$A - \lambda I = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 1 \\ 3 & 2 & 3 \end{bmatrix} - \lambda \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 - \lambda & 1 & 1 \\ 1 & 2 - \lambda & 1 \\ 3 & 2 & 3 - \lambda \end{bmatrix}$$ Now, compute the determinant of $$A - \lambda I$$: $$\det(A - \lambda I) = \det \begin{bmatrix} 1 - \lambda & 1 & 1 \\ 1 & 2 - \lambda & 1 \\ 3 & 2 & 3 - \lambda \end{bmatrix}$$ Let's calculate this determinant. I'll perform the computation step by step. ### Step 2: Solve the Characteristic Polynomial Let me calculate the determinant and then solve for the characteristic polynomial.The characteristic polynomial of the matrix $$A$$ is: $$p(\lambda) = \lambda(-\lambda^2 + 6\lambda - 5) = -\lambda^3 + 6\lambda^2 - 5\lambda$$ ### Step 3: Apply the Cayley-Hamilton Theorem According to the Cayley-Hamilton theorem, the matrix $$A$$ must satisfy its characteristic equation, which is obtained by replacing $$\lambda$$ in the characteristic polynomial with $$A$$: $$p(A) = -A^3 + 6A^2 - 5A = 0$$ Now, let's calculate $$A^3$$, $$A^2$$, and then verify the equation $$-A^3 + 6A^2 - 5A = 0$$. I'll compute these matrices step by step.After computing the expression $$-A^3 + 6A^2 - 5A$$, the result is the zero matrix, confirming that: $$-A^3 + 6A^2 - 5A = 0$$ ### Conclusion: The matrix $$A$$ satisfies its characteristic equation, thus verifying the Cayley-Hamilton theorem for the given matrix. --- Would you like more details on any step or further clarification? Here are some related questions you may find useful: 1. How is the characteristic polynomial derived for larger matrices? 2. What are the applications of the Cayley-Hamilton theorem in solving linear systems? 3. Can the Cayley-Hamilton theorem be used to compute matrix functions, such as exponentials? 4. How can we compute the inverse of a matrix using the Cayley-Hamilton theorem? 5. What is the significance of eigenvalues in verifying the Cayley-Hamilton theorem? **Tip:** The Cayley-Hamilton theorem is a powerful tool that simplifies matrix computations by reducing higher powers of matrices to linear combinations of lower powers.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Theory
Characteristic Polynomial
Formulas
Characteristic Polynomial: p(λ) = det(A - λI)
Cayley-Hamilton Theorem: A satisfies its own characteristic equation p(A) = 0
Theorems
Cayley-Hamilton Theorem
Suitable Grade Level
Undergraduate Level (Linear Algebra Course)
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