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-2 1. Suppose G = -4 Determine the values of x for which det(G) = 0. -4 2 x + 4,

-2 1. Suppose G = -4 Determine the values of x for which det(G) = 0. -4 2 x + 4,

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Please answer question #2 with details on how to do it.

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### Linear Algebra: Suggested Exercises Using Determinants and the Cayley-Hamilton Theorem #### Exercise 1 Suppose \( G = \begin{pmatrix} -2 & 1 & 5 \\ 0 & x & -4 \\ -4 & 2 & x + 4 \end{pmatrix} \). Determine the values of \( x \) for which \( \text{det}(G) = 0 \). #### Exercise 2 Suppose that \( Z = \begin{pmatrix} 0 & 15 & 0 \\ 2 & 4 & 0 \\ 3 & 6 & 6 \end{pmatrix} \). Given that the characteristic polynomial of this matrix is \( P(\lambda) = \lambda^3 - 20\lambda^2 + 94\lambda - 60 \), use the Cayley-Hamilton theorem to find \( Z^{-1} \). #### Exercise 3 Suppose that \( K = \begin{pmatrix} 2 & -1 \\ 0 & 5 \end{pmatrix} \). Use the Cayley-Hamilton theorem to find \( K^3 \). ### Explanation of Key Concepts: 1. **Determinant (det)**: - The determinant is a scalar value that is computed from the elements of a square matrix. - It is used in a variety of matrix-related problems, including determining whether a matrix is invertible. 2. **Characteristic Polynomial**: - The characteristic polynomial of a matrix is obtained by computing the determinant of \(\lambda I - A\), where \(I\) is the identity matrix and \(A\) is the given matrix. - It is denoted as \( P(\lambda) \) and is used in the Cayley-Hamilton theorem. 3. **Cayley-Hamilton Theorem**: - This theorem states that every square matrix over a commutative ring (like real or complex numbers) satisfies its own characteristic polynomial. - For a matrix \( A \) with characteristic polynomial \( P(\lambda) = \lambda^n + c_{n-1}\lambda^{n-1} + \cdots + c_1 \lambda + c_0 \), the theorem asserts that \( P(A) = A^n + c_{n-1}A^{n-1} + \cdots
Transcribed Image Text:### Linear Algebra: Suggested Exercises Using Determinants and the Cayley-Hamilton Theorem #### Exercise 1 Suppose \( G = \begin{pmatrix} -2 & 1 & 5 \\ 0 & x & -4 \\ -4 & 2 & x + 4 \end{pmatrix} \). Determine the values of \( x \) for which \( \text{det}(G) = 0 \). #### Exercise 2 Suppose that \( Z = \begin{pmatrix} 0 & 15 & 0 \\ 2 & 4 & 0 \\ 3 & 6 & 6 \end{pmatrix} \). Given that the characteristic polynomial of this matrix is \( P(\lambda) = \lambda^3 - 20\lambda^2 + 94\lambda - 60 \), use the Cayley-Hamilton theorem to find \( Z^{-1} \). #### Exercise 3 Suppose that \( K = \begin{pmatrix} 2 & -1 \\ 0 & 5 \end{pmatrix} \). Use the Cayley-Hamilton theorem to find \( K^3 \). ### Explanation of Key Concepts: 1. **Determinant (det)**: - The determinant is a scalar value that is computed from the elements of a square matrix. - It is used in a variety of matrix-related problems, including determining whether a matrix is invertible. 2. **Characteristic Polynomial**: - The characteristic polynomial of a matrix is obtained by computing the determinant of \(\lambda I - A\), where \(I\) is the identity matrix and \(A\) is the given matrix. - It is denoted as \( P(\lambda) \) and is used in the Cayley-Hamilton theorem. 3. **Cayley-Hamilton Theorem**: - This theorem states that every square matrix over a commutative ring (like real or complex numbers) satisfies its own characteristic polynomial. - For a matrix \( A \) with characteristic polynomial \( P(\lambda) = \lambda^n + c_{n-1}\lambda^{n-1} + \cdots + c_1 \lambda + c_0 \), the theorem asserts that \( P(A) = A^n + c_{n-1}A^{n-1} + \cdots
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