The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of the matrix shown below is as follows. 2 - 61 + 11 = 0 and by the theorem you have A2 - 6A + 11I, - 0 1 -3 A = 2 5 Demonstrate the Cayley-Hamilton Theorem for the matrix A given below. 05 -1 A = -15 -1 0 0 -1 STEP 1: Find and expand the characteristic equation. STEP 2: Compute the required powers of A. A = STEP 3: Write a matrix version of the characteristic equation by replacing a with A. (Use I for the 3x3 identity matrix.) STEP 4: Substitute the powers of A into the matrix equation from step 3, and simplify. Is the matrix equation true? O Yes O No
The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of the matrix shown below is as follows. 2 - 61 + 11 = 0 and by the theorem you have A2 - 6A + 11I, - 0 1 -3 A = 2 5 Demonstrate the Cayley-Hamilton Theorem for the matrix A given below. 05 -1 A = -15 -1 0 0 -1 STEP 1: Find and expand the characteristic equation. STEP 2: Compute the required powers of A. A = STEP 3: Write a matrix version of the characteristic equation by replacing a with A. (Use I for the 3x3 identity matrix.) STEP 4: Substitute the powers of A into the matrix equation from step 3, and simplify. Is the matrix equation true? O Yes O No
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.1: Matrix Operations
Problem 20EQ: Referring to Exercise 19, suppose that the unit cost of distributing the products to stores is the...
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![### Demonstrating the Cayley-Hamilton Theorem
The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of the matrix shown below is as follows:
\[
A = \begin{bmatrix}
1 & -3 \\
2 & 5
\end{bmatrix}
\]
\[
\lambda^2 - 6\lambda + 11 = 0
\]
and by the theorem you have
\[
A^2 - 6A + 11I = 0
\]
### Demonstrate the Cayley-Hamilton Theorem for the matrix \( A \) given below.
\[
A = \begin{bmatrix}
0 & 5 & -1 \\
-1 & 5 & -1 \\
0 & 0 & -1
\end{bmatrix}
\]
### STEP 1:
**Find and expand the characteristic equation.**
*Input area for characteristic equation expansion*
### STEP 2:
**Compute the required powers of \( A \).**
\[
A^2 = \begin{bmatrix}
\_\_ & \_\_ & \_\_ \\
\_\_ & \_\_ & \_\_ \\
\_\_ & \_\_ & \_\_
\end{bmatrix}
\]
\[
A^3 = \begin{bmatrix}
\_\_ & \_\_ & \_\_ \\
\_\_ & \_\_ & \_\_ \\
\_\_ & \_\_ & \_\_
\end{bmatrix}
\]
### STEP 3:
**Write a matrix version of the characteristic equation by replacing \( \lambda \) with \( A \). (Use \( I \) for the 3x3 identity matrix.)**
*Input area for matrix version of characteristic equation*
### STEP 4:
**Substitute the powers of \( A \) into the matrix equation from step 3, and simplify. Is the matrix equation true?**
*Yes*
*No*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F42f67b29-db3a-470e-8f57-9956af53c593%2Fb2c91950-ee5c-44fe-ac83-63a3a0b79f0f%2Fl7uy1yk_processed.png&w=3840&q=75)
Transcribed Image Text:### Demonstrating the Cayley-Hamilton Theorem
The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of the matrix shown below is as follows:
\[
A = \begin{bmatrix}
1 & -3 \\
2 & 5
\end{bmatrix}
\]
\[
\lambda^2 - 6\lambda + 11 = 0
\]
and by the theorem you have
\[
A^2 - 6A + 11I = 0
\]
### Demonstrate the Cayley-Hamilton Theorem for the matrix \( A \) given below.
\[
A = \begin{bmatrix}
0 & 5 & -1 \\
-1 & 5 & -1 \\
0 & 0 & -1
\end{bmatrix}
\]
### STEP 1:
**Find and expand the characteristic equation.**
*Input area for characteristic equation expansion*
### STEP 2:
**Compute the required powers of \( A \).**
\[
A^2 = \begin{bmatrix}
\_\_ & \_\_ & \_\_ \\
\_\_ & \_\_ & \_\_ \\
\_\_ & \_\_ & \_\_
\end{bmatrix}
\]
\[
A^3 = \begin{bmatrix}
\_\_ & \_\_ & \_\_ \\
\_\_ & \_\_ & \_\_ \\
\_\_ & \_\_ & \_\_
\end{bmatrix}
\]
### STEP 3:
**Write a matrix version of the characteristic equation by replacing \( \lambda \) with \( A \). (Use \( I \) for the 3x3 identity matrix.)**
*Input area for matrix version of characteristic equation*
### STEP 4:
**Substitute the powers of \( A \) into the matrix equation from step 3, and simplify. Is the matrix equation true?**
*Yes*
*No*
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