Practice Problems

MTH 2201

1. Find the eigenvalues and eigen vectors of

i) A =

[10 −94 −2

]. Solution: (i) λ = 4, 4; X = t

[32

]

(ii) A =

3 4 −1−1 −2 13 9 0

Solution: (i) λ1 = 2, 2; X = t

−113

; λ2 = −3; X = t

−11−2

.

2. Find the eigen values and eigen vectors of A and of the stated power of A.

(i) A =

−1 −2 −21 2 1−1 −1 0

; A25

Solution: λ1 = −1; X = t

2−1

1

; λ2 = 1; X = t

0−1

1

+ s

1−1

0

.

The eigen values of A25 are λ = (−1)25 = −1 and λ2 = (1)25 = 1.

3. For what value(s) of x if any does the matrix A =

3 0 00 x 20 2 x

, has atleast

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 andx− 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 characteristicpolynomial.

6. Let A =

[2 02 3

]. Show that A and AT do not have the same eigen spaces.

7. If λ is an eigen value of A and X is the corresponding eigenvector, then provethat λ− s is an eigen value of A− sI for any scalar s and X is thecorresponding eigenvector.

8. Let A =

[2 02 3

]and B1, B2, B3 be the matrices obtained by the elementary

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 thematrix A + 3I ?

2