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 April 28th, 2009, 07:02 AM #1 Newbie   Joined: Apr 2009 Posts: 28 Thanks: 0 Matrix Problem Hi, I need some help to solve the following problem. Let A be any 2 x 2 matrix over C and let f(x) = a0 + a1x + a2.x^2 + ..... + an.x^n be any polynomial over the complex numbers C. Show that f(A) is a matrix which can be written as c1.I + c2.A for some c1, c2 belongs to C, where I is the identity matrix. Can somebody help me how to start it? Thanks in advance!
 April 28th, 2009, 10:13 AM #2 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 Re: Matrix Problem If you can show it for any $g(x)=x^k,\quad k\in\mathbb{N}$ then you can show it for an arbitrary polynomial $f.$ Then, show that it is possible to write $A^2$ as $c_1I+c_2A$, and then show by induction that it is possible for any $A^k.$
 April 28th, 2009, 07:57 PM #3 Senior Member   Joined: Jul 2008 Posts: 144 Thanks: 0 Re: Matrix Problem the degree of minimal polynomial of $A$ is less then 2. so in polynominal ring $f(A)\in C[A],deg(f(A))\leq1$
April 29th, 2009, 05:49 AM   #4
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Re: Matrix Problem

Quote:
 Originally Posted by mattpi ...show that it is possible to write $A^2$ as $c_1I+c_2A$...
Hint: consider the expression $A^2-(\operatorname{Tr} A)A+(\det A)I$ for an arbitrary 2x2 matrix $A.$

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