For the matrix A below compute A999 by hand ie dont just put

For the matrix, A below, compute A^999 by hand. (i.e. don\'t just put it in a calculator - instead, make use of eigenvalues and ...) A = (1 2 3 4 0 5 6 7 0 0 8 9 0 0 0 10)

The characteristic equation of A is det (A- I₄) = 0 or, (-10)( -8)( -5)( -1) = 0. Tus, the eigenvalues of A are ₁ =10 , ₂ = 8 ₃ =5 and ₄ =1 The eigenvectors of A, corresponding to the eigenvalue are solutions to the equation (A- I₄)X = 0. These are v₁ = (311,612,405,90)^T ,v₂ = (1,2,1,0)^T , v₃ = ( 1,2,0,0)^T and v₄ = (1,0,0,0)^T. Then A = PDP^-1, where P =

311

612

405

and P^-1 =

1/90

-9/2

1/2

-1

11/10

-1/2

-1/18

Finally, A⁹⁹⁹ = PD⁹⁹⁹P^-1 where D⁹⁹⁹ =

10⁹⁹⁹

8⁹⁹⁹

5⁹⁹⁹

311	1	1	1
612	2	2	0
405	1	0	0
90	0	0	0

$For the matrix, A below, compute A^999 by hand. (i.e. don\'t just put it in a calculator - instead, make use of eigenvalues and ...) A = (1 2 3 4 0 5 6 7 0 0 8$

For the matrix A below compute A999 by hand ie dont just put

Solution

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