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Given a Matrix A = [a,b;c,d] and it's characteristic polynomial, why does the characteristic polynomial enables us to determine the result of the Matrix A raised to the nth power?
The Cayley Hamilton Theorem is a fundamental theorem in linear algebra that states that every square matrix satisfies its own characteristic equation. In simpler terms, this means that a square matrix can be used to find its own eigenvalues.
The Cayley Hamilton Theorem was first proved by the mathematician Arthur Cayley in 1858. However, it was later generalized by the mathematician William Hamilton in 1859.
The Cayley Hamilton Theorem has many applications in fields such as physics, engineering, and economics. It is used to solve systems of linear differential equations, analyze stability of dynamical systems, and study the behavior of matrices in various physical systems.
Yes, the Cayley Hamilton Theorem only applies to square matrices. This is because the characteristic equation, which is used to find eigenvalues, can only be defined for square matrices.
No, the Cayley Hamilton Theorem only applies to matrices in commutative rings, where the order of multiplication does not affect the result. In non-commutative rings, the theorem does not hold and there may not even be a characteristic equation for a given matrix.