MATH 218 Problems, Spring 2011

Extra Problems 1. For which values of a and b does the following system has no solution exactly one or infinitely many solutions:      ax + z = 2 ax + ay + 4z = −4 (a − 1)y + 2z = b 2. For which value(s) of a is the set S = {ax3, x3 + (2a − 1)x2, a2, x2 + (a − 4)x} linearly inde- pendent. 3. a- Find matrices A, B, C such that AC = BC with A ̸= B. b-Determine all 2 × 2 matrices A such that A2 = 0. 4. For which values of x and y is the matrix A invertible: A =      x y 0 0 0 x y 0 0 0 x y y 0 0 x      5. If (I + (2A)−1)−1 = � 1 −2 −3 5 � . Find A. 6. Let A be a square matrix such that : A3 + 4AAT − 2A − 7I = 0. Find an expression for A−1. 7. Let J =      1 −1 0 0 −1 1 0 0 0 0 1 1 0 0 0 1     . Find J2 and J3. Deduce Jn. 8. Let Jn be the n × n matrix each of which entries is 1. Show that if n > 1 then: (I − Jn)−1 = I − 1 n − 1J 9. True/False (prove or explain if true, give a counter-example if false) a- If ATA is invertible so is A. b- det(A + B) = det(A) + det(B). c- If A is invertible then Ax = x has exactly one solution. d. If {v1, v2, . . . vr} is a linearly dependent set then {v1, v2, . . . vr, vr+1 . . . vn} is also linearly dependent. 10. p : 90 : 8 − 9 − 10 review chapter test p : 90 : 2, 3, 7, 10, 15, 17, 19, 25