MATH 218 Exam 1 with Solution, Fall 2005

AMERICAN UNIVERSITY OF BEIRUT Mathematics Department Math 218 - Quiz I Fall 2005-2006 Section 1 Section 2 Section 3 Mrs. Z. Mouzeihem Ms. M. Houri Ms. D. Audi Time: 60 min I- (20 points) For what values of m does the following system have: x + mz 0 ; {: +y A] : 6) xz +my — 2z -1 i- no solutions. atl iO ii- infinitely many solutions. ij- a unique solution. 12 0 A=|]2 2 4 121 a- (10 points) Find the inverse of the matrix A. b- (5 points) Use part (a) to solve the following system a + 2y = 2 25 +2y +42 =-1 gz +2y +z =2 — Xie Ab < IIJ- a (5 points) Let A be a square matrix such that A® + 4AAT — 2A —7I = 0. Show that A is invertible and find A7!. A (At Aa AT. 2) = [a (atea®e ye L poe aD) -— 7 b- (5 points) Let A, B and C be square matrices of the same size such that B = A~!C.A. Show that B? = A-!C?A and deduce B”. e-pca gee atica ACA - ic A a7 NE] sich gost IV- (10 points) Find the values of k for which the matrix is not invertible. A- wins (1 12) a- (10 points) Find det(A) = - 3 ne 3( 2) = 3 , b- (20 points) Find the following i- det(A*) oa J ~3) . . (2A eH z oe (Ali iii- det(A? A71) iv- det{adj(A)} — = b K A. ad J) = TAI 1 Lad; A) = Zs (Al VI- a- (10 points) Answer TRUE or FALSE without proof. i- (det(A))? = det(A?) 7 ii- If det(A) = 0 the system AX = b has infinitely many solutions. -F iii- Let A and B be two matrices of the same size, if AB is invertible then A and B are both invertible. = 77 . iv- If A is invertible then the system AX = 2X has exactly one f. solution. v- tr(A?)=tr(A) T b- (5 points) Justify your answer if TRUE or give a counter example if FALSE for part (iii) ONLY. (AL 14 fe pAb GOOD LUCK!