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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