'Verify the Cayley Hamilton theorem for the matrix 3)'
Added by Alejandro B.
Step 1
The characteristic polynomial of a matrix is obtained by finding the determinant of the matrix (λI - A), where λ is a scalar and I is the identity matrix of the same size as A. For the given matrix 3, we have: (λI - A) = (λ - 3) det(λI - A) = det(λ - 3) = (λ - Show more…
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Cayley-Hamilton Theorem In Exercises $49-52$ demonstrate the Cayley-Hamilton Theorem for the matrix $A$. The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of $A=\left[\begin{array}{rr}1 & -3 \\ 2 & 5\end{array}\right]$ is $\lambda^{2}-6 \lambda+11=0,$ and by the theorem you have $A^{2}-6 A+11 I_{2}=O$. $$A=\left[\begin{array}{rrr}-3 & 1 & 0 \\-1 & 3 & 2 \\0 & 4 & 3\end{array}\right]$$
Eigenvalues and Eigenvectors
Cayley-Hamilton Theorem In Exercises $49-52$ demonstrate the Cayley-Hamilton Theorem for the matrix $A$. The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of $A=\left[\begin{array}{rr}1 & -3 \\ 2 & 5\end{array}\right]$ is $\lambda^{2}-6 \lambda+11=0,$ and by the theorem you have $A^{2}-6 A+11 I_{2}=O$. $$A=\left[\begin{array}{rrr}1 & 0 & -4 \\0 & 3 & 1 \\2 & 0 & 1\end{array}\right]$$
Cayley-Hamilton Theorem In Exercises $49-52$ demonstrate the Cayley-Hamilton Theorem for the matrix $A$. The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of $A=\left[\begin{array}{rr}1 & -3 \\ 2 & 5\end{array}\right]$ is $\lambda^{2}-6 \lambda+11=0,$ and by the theorem you have $A^{2}-6 A+11 I_{2}=O$. $$A=\left[\begin{array}{rr}5 & 0 \\-7 & 3\end{array}\right]$$
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