Re: Linear Algebra

In article <16059674.1174160244046.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
Eric <starwar636@xxxxxxx> wrote:
If I have a matrix A =[7 0 0;0 1 2;0 1 -1] (Matlab notation), how can I express the inverse of A (A^-1) as a polynomial of A with real coefficients.


Also, if M^3=I but M is not equal to I, how can I find the eigenvalues of M?

The characteristic polynomial of A is:

- lambda^3 + 7 lambda^2 + 3 lambda - 21

so we have:

A^3 - 7 A^2 - 3A + 21 I = 0

A^2 - 7A - 3I = -21 A^{-1}

A^{-1} = -A^2/21 + A/3 + I/7

Since, if v is an eigvenvector of M, we have M^3 v = lambda^3 v = v,
the eigenvalues of M must be roots of lambda^3 - 1 = 0.

--
Daniel Mayost

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