My Math Forum Show that A must be invertible

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 November 27th, 2009, 02:39 PM #1 Senior Member   Joined: Nov 2009 Posts: 169 Thanks: 0 Show that A must be invertible Let $P(x)= a0 + a1x + a2x^2 + ... + anx^n$ be a polynomial of degree n in x with real coefficients. For any mxm matrix A, we define $P(A)= a0Im + a1A + a2A^2 + ... + anA^n$. Show that, if $P(A)= 0m$and $a0 not= 0$, then A must be invertible. [Hint: isolate Im from the equation P(A) = 0m] For A = [2 2 -1] [-1 -1 1] [2 4 -1] , compute $A^2$, if $P(x)= x^2 +x -2$, calculate P(A). ) Can you explan this,>? For p(A), How can I subtract 2 from a mtrix?
 November 28th, 2009, 01:32 AM #2 Senior Member   Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 Re: Show that A must be invertible Subtracting 2 from a matrix is really subtracting 2I... that is $2I_3 = \left[\begin{array} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2\end{array}\right]$ P(A) (as a formula on matrices) will be A^2 + A - 2I Does that answer your question or is there something I missed? Also, in latex, you can get subscripts as well; just but _ after the letter. E.g. a_0 gives you $a_0$ If you need more than one symbol in the subscript, wrap it in brackets. So A_{0,0} gives you $A_{0,0}$

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