Solved: Suppose A is diagonalizable and p (t) is the

Chapter , Problem 7E

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Problem 7E

Suppose A is diagonalizable and p (t) is the characteristic polynomial of A. Define p (A) as in Exercise 5, and show that p (A) is the zero matrix. This fact, which is also true for any square matrix, is called the Cayley–Hamilton theorem.

Reference:

If  p(t) = c0 + c1t + c2t2 + + cntn, define p (A)

to be the matrix formed by replacing each power of t in p (t) by the corresponding power of A (with A0 = I ). That is,

p (A) = c0I + c1A + c2A2 + + cnAn

Show that if   is an eigenvalue of A, then one eigenvalue of

p (A) is p ().