Practice Problems
MTH 2201
1. Find the
" eigenvalues
# and eigen vectors of
"
#
10 −9
3
i) A =
. Solution: (i) λ = 4, 4; X = t
4 −2
2


3
4 −1
1 
(ii) A = 
 −1 −2

3
9
0




−1
−1



Solution: (i) λ1 = 2, 2; X = t 
 1  ; λ2 = −3; X = t  1  .
3
−2
2. Find the eigen values and
 eigen vectors of A and of the stated power of A.
−1 −2 −2

2
1 
(i) A =  1
 ; A25
−1 −1 0






2
0
1






Solution: λ1 = −1; X = t  −1  ; λ2 = 1; X = t  −1  + s  −1  .
1
1
0
25
25
25
The eigen values of A are λ = (−1) = −1 and λ2 = (1) = 1.


3 0 0


3. For what value(s) of x if any does the matrix A =  0 x 2  , has atleast
0 2 x
one repeated eigenvalue. (solution: x = 1 or x = 5.)
4. Let A be a (2 × 2) matrix such that A2 = I. For any x ∈ IR2 , if x + Ax and
x − Ax are eigenvectors of A find the corresponding eigenvalue.
5. Prove that if A is a square matrix then A and AT have the same characteristic
polynomial.
"
6. Let A =
#
2 0
. Show that A and AT do not have the same eigen spaces.
2 3
7. If λ is an eigen value of A and X is the corresponding eigenvector, then prove
that λ − s is an eigen value of A − sI for any scalar s and X is the
corresponding eigenvector.
"
#
2 0
8. Let A =
and B1 , B2 , B3 be the matrices obtained by the elementary
2 3
row operations R2 → R2 − R1 , R2 ↔ R1 and R2 → (−2)R2 respectively on A.
Find the eigen values of A, B1 , B2 and B3 .
1
9. Do you observe any relation between the eigenvalues of the matrices A, B1
and A + B1 .
10. What is the relation between the eigen values of a matrix A and those of the
matrix A + 3I ?
2