MA2503 – Exercises for Week 5
1. Find the eigenvalues and corresponding eigenvectors (or bases for the eigenspaces) of the
matrix. Is the matrix diagonalizable? If it is, write down the invertible matrix P and the
diagonal matrix D, such that P −1 AP = D.




1 0 −2
6 3 −8




(a).  0 0 0  (b). 0 −2 0 
−2 0 4
1 0 −3
2. Find all possible real 3 × 3 matrix A, such that A2 (yes, it is A2 , not A) has eigenvalues
1, 1 and 0, for which
     
1
1
1
     
−1 , 1 , −1 ,
1
0
0
are their corresponding eigenvectors (of A2 ).
3. If A is a 2 × 2 matrix, λ1 and λ2 are the two eigenvalues of A, p(λ) = det(λI − A) =
λ2 + c1 λ + c2 is the characteristic polynomial, show that
p(λ) = λ2 − trace(A)λ + det(A)
det(A) = λ1 λ2
trace(A) = λ1 + λ2
p(A) = A2 + c1 A + c2 I = 0.
4. For matrix A below, find A1000 , A−1000 , A2301 , A−2301 .


1 −2 8


A = 0 −1 0 
0 0 −1