Summer 2017 Quiz 2 A Key.pdf - Laboratory Quiz#2A Time 20 minutes Name Student ID Answer all questions in the provided space No calculators

# Summer 2017 Quiz 2 A Key.pdf - Laboratory Quiz#2A Time 20...

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Laboratory Quiz #2A Time: 20 minutes Name: Student ID Answer all questions in the provided space. No calculators allowed. ( Total Marks: 25) . 1. The following system of equations can be written as Ax=b. x - z = - 3 x + y = 2 - 2 x + 3 z = 4 (a) [5] Find A - 1 Solution: 1 0 - 1 1 0 0 1 1 0 0 1 0 - 2 0 3 0 0 1 1 0 - 1 1 0 0 0 1 1 - 1 1 0 - 2 0 3 0 0 1 R 2 - R 1 R 2 1 0 - 1 1 0 0 0 1 1 - 1 1 0 0 0 1 2 0 1 R 3 + 2 R 1 R 3 1 0 - 1 1 0 0 0 1 0 - 3 1 - 1 0 0 1 2 0 1 R 2 - R 3 R 2 1 0 0 3 0 1 0 1 0 - 3 1 - 1 0 0 1 2 0 1 R 1 + R 3 R 1 A - 1 = 3 0 1 - 3 1 - 1 2 0 1 (b) [3] Use A - 1 to solve Ax = b Solution: x = A - 1 b = 3 0 1 - 3 1 - 1 2 0 1 3 2 4 = 13 - 11 10 1
2. The matrixA=1-112can be reduced down to the identity in the following three steps:ARowOp11-103RowOp21-101RowOp31001= I(a)[3]What are the three elementary row operations used? (b)[3]Using part (a), find elementary matricesE1,E2andE3such thatE3E2E1A= I. (c)[3]Find elementary matricesE-11,E-12andE-13such thatA=E-11E-12E-13. R2+R1R2E-12=13,(3R2R2E-13=1-11,R1-R2R12
3.[6]LetA=Use cofactor expansionalong the3rdrow, in every step, to evaluate the determinant ofA.692241-1 69322411-1. Solution:-1=-1(1)6924+ (-1)622=-1[24-18-(-12)]=-1[24-6]=-18 4.[2]Prove that ifAis a symmetric matrix, thenkAis symmetric. Solution:By assumption,AT=A. Therefore, (kA)T=kAT=kA. 3