The Cayley-Hamilton Theorem provides a method for calculating powers of a matrix. For example, if A is a 3x 3 matrix with the characteristic equation Co + Cz1 +C212 + 13 = 0 then col + c,A + C2A² + A³ = 0, so A3 = - C2A? - C1A – col Multiplying through by A yields A4 = - coA – c¿A² – C2A³, which expresses A“ in terms of A3, A² and A. Use this procedure to calculate A3 and A4 for 0. 1 0 A =| 0 0 1 1 -4 4
The Cayley-Hamilton Theorem provides a method for calculating powers of a matrix. For example, if A is a 3x 3 matrix with the characteristic equation Co + Cz1 +C212 + 13 = 0 then col + c,A + C2A² + A³ = 0, so A3 = - C2A? - C1A – col Multiplying through by A yields A4 = - coA – c¿A² – C2A³, which expresses A“ in terms of A3, A² and A. Use this procedure to calculate A3 and A4 for 0. 1 0 A =| 0 0 1 1 -4 4
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![The Cayley-Hamilton Theorem provides a method for calculating powers of a matrix. For example, if \( A \) is a \( 3 \times 3 \) matrix with the characteristic equation
\[ c_0 + c_1 \lambda + c_2 \lambda^2 + \lambda^3 = 0 \]
then \( c_0I + c_1A + c_2A^2 + A^3 = 0 \), so
\[ A^3 = -c_2A^2 - c_1A - c_0I \]
Multiplying through by \( A \) yields
\[ A^4 = -c_0A - c_1A^2 - c_2A^3 \]
which expresses \( A^4 \) in terms of \( A^3, A^2 \), and \( A \). Use this procedure to calculate \( A^3 \) and \( A^4 \) for
\[ A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & -4 & 4 \end{bmatrix} \]
\( A^3 \) is
\[
\begin{bmatrix}
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{}
\end{bmatrix}
\]
\( A^4 \) is
\[
\begin{bmatrix}
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{}
\end{bmatrix}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9b53ebc-804c-49f3-b917-c4e1c4975b52%2F5cdcf2f0-de36-4d11-9d99-391acf963446%2Fowz93d_processed.png&w=3840&q=75)
Transcribed Image Text:The Cayley-Hamilton Theorem provides a method for calculating powers of a matrix. For example, if \( A \) is a \( 3 \times 3 \) matrix with the characteristic equation
\[ c_0 + c_1 \lambda + c_2 \lambda^2 + \lambda^3 = 0 \]
then \( c_0I + c_1A + c_2A^2 + A^3 = 0 \), so
\[ A^3 = -c_2A^2 - c_1A - c_0I \]
Multiplying through by \( A \) yields
\[ A^4 = -c_0A - c_1A^2 - c_2A^3 \]
which expresses \( A^4 \) in terms of \( A^3, A^2 \), and \( A \). Use this procedure to calculate \( A^3 \) and \( A^4 \) for
\[ A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & -4 & 4 \end{bmatrix} \]
\( A^3 \) is
\[
\begin{bmatrix}
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{}
\end{bmatrix}
\]
\( A^4 \) is
\[
\begin{bmatrix}
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{}
\end{bmatrix}
\]
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