Inverse of A using Cayley-Hamilton?

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In summary, The Cayley-Hamilton theorem is a useful tool in finding the inverse of a square matrix. It states that every square matrix satisfies its own characteristic polynomial, which can be used to calculate the inverse of the matrix. To find the inverse of a matrix using this theorem, the characteristic polynomial is first calculated and then used in a specific formula. However, this method is limited to square matrices and can be computationally intensive for larger matrices. It cannot be used to find the inverse of a singular matrix, as a singular matrix does not have an inverse.
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Rebbeca
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Homework Statement


   2 -1 1
A = -1 2 -1
   -1 -1 2
Find A^-1 using Cayley Hamilton Theorem?

Homework Equations


The Attempt at a Solution



http://i.imm.io/lyhB.jpeg [Broken]I came thus far,

0 = -A^3 + 6A^2 - 11 + 6I

How do I manipulate the above to get A^-1?
 
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  • #2
Multiply by A^(-1) and simplify?
 

What is the Inverse of A using Cayley-Hamilton?

The inverse of a matrix A can be found using the Cayley-Hamilton theorem, which states that every square matrix satisfies its own characteristic polynomial. The characteristic polynomial can be used to calculate the inverse of A.

What is the Cayley-Hamilton Theorem?

The Cayley-Hamilton theorem is a mathematical theorem that states every square matrix satisfies its own characteristic polynomial. This theorem is useful in finding the inverse of a matrix.

How do you use the Cayley-Hamilton Theorem to find the Inverse of A?

To find the inverse of a matrix A using the Cayley-Hamilton theorem, you first need to calculate the characteristic polynomial of A. Then, use the characteristic polynomial to find the inverse of A using the formula (A - λI)^n = 0, where n is the dimension of the matrix and λ is the eigenvalue of A.

What are the limitations of using Cayley-Hamilton to find the Inverse of A?

While the Cayley-Hamilton theorem is a useful tool in finding the inverse of a matrix, it is limited to square matrices. Additionally, it can be computationally intensive for larger matrices.

Can Cayley-Hamilton be used to find the Inverse of a singular matrix?

No, the Cayley-Hamilton theorem cannot be used to find the inverse of a singular matrix. This is because a singular matrix does not have an inverse. In fact, the characteristic polynomial of a singular matrix is equal to 0, making it impossible to use the Cayley-Hamilton theorem.

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