Questions:
Fundamental Mathematics: Eigenvectors and Matrices
1) Find the eigenvectors and eigenvalues of the matrix A. Hence find the matrix P and a diagonal matrix D such that A = P^1 DP and compute A^100
Where
i)A = (7 -6)
(8 -7)
( 2 0 -3)
ii) A = ( 1 1 -5)
( 0 0 1)
2.i) Verify the Cayley-Hamilton theorem for the matrix
A =( a b)
( c d)
ii) Compute the minimal polynomial of a matrix
( 0 -1 1)
A = ( 1 2 -1)
( 1 1 0)
and decide whether the matrix is diagonalizable or not. The same question for the matrix
( 1 1 0 0)
B = ( -1 -1 0 0)
( -2 -2 2 1)
( 1 1 -1 0)