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Diﬀerential Equations, Math. 308–200. Spring 2011. Instructor P. Kuchment. Assignment A5. Due March 10th. 18 points (1 point for each problem). Show all work. 1. Find the matrix of the following system and write the system in the vector form: { 3𝑥1 − 2𝑥2 = 5 2𝑥2 + 𝑥1 = 7 ( 2. 3. 4. 5. 6. ) 3 Compute 𝐴x, where 𝐴 = ,x = . 1 ( ) ( ) 2 −1 3 5 Compute 𝐴𝐵 and 𝐵𝐴, where 𝐴 = ,𝐵 = . 𝜋 0 1 2 ( ) 3 1 The matrix 𝐴 = creates a linear mapping (transformation) 1 1 of the plane (into )itself. What ( are) the images of the standard basis 1 0 vectors 𝑒1 = and 𝑒2 = ? 0 1 ( ) 2 For the same basis vectors, ﬁnd a matrix 𝐴 such that 𝐴𝑒1 = −2 ( ) −3 and 𝐴𝑒2 = . Is such a matrix unique? 1 ( ) 2 −1 Find the transpose 𝐴𝑡 to the matrix 𝐴 = 3 1 2 −1 𝜋 0 ) ( 7. Compute det 𝐴 and tr𝐴 for the matrix in the previous exercise. ( ) −2 8. What is the area of the parallelogram that has vectors 𝑎 = 2 ( ) 3 and 𝑏 = as its sides? 1 9. A matrix 𝐴 sends the standard basis vectors 𝑒1 , 𝑒2 (see Problem 4) into two parallel (collinear) vectors. What is the determinant of 𝐴? 1 10. What does it mean that a matrix is non-singular? Write examples of a non-singular and of a singular matrices (with non-zero entries). 11. Find the inverse for the matrix 𝐴 in Problem 4. 12. Use the result of the previous problem and matrix algebra to solve the system { 3𝑥1 + 𝑥2 = 5 𝑥1 + 𝑥2 = 7 13. Let 𝜆1 , 𝜆2 be the eigenvalues of a matrix 𝐴. Prove that 𝜆1 + 𝜆2 = tr𝐴, 𝜆1 × 𝜆2 = det 𝐴. 14. Can a matrix 𝐴 with real entries have eigenvalues 𝜆1 = 2 + 𝑖, 𝜆2 = 3? 15. Construct an example of a matrix that has eigenvalues 𝜆1 = 2, 𝜆2 = 3. Find eigenvalues and corresponding eigenvectors of the following matrices: ( ) 5 3 16. 𝐴 = −4 −2 ( ) 4 5 17. 𝐴 = −5 −6 ( ) 3 1 18. 𝐴 = −2 1 2