Unformatted text preview: MA405H, TEST 1 Department of Mathematics, NCSU SHOW YOUR WORK. NO WORK = NO CREDIT.
NO CALCULATORS ARE ALLOWED ’ 1. (20%)C0nsider the linear system :01 + 2x2 — 3:223 = 4
3.1171 — $2 + 53153 = 2
4951 + 332 + (a2 —14)x3 a + 2 Find all the values of ‘a’ for which this system will have:
(i) no solution, (ii) exactly one solution, or (iii) inﬁnitely many solutions. 2. (10%) Consider the n X n homogeneous linear system AX = 0 and suppose that X1 aé 0 is a
solution. (a) How many solutions will this linear system have? Justify your answer. (b) Is the matrix A invertible? Justify your answer. 3. Let (a) 10%) Find the LU—decomposition of A.
(b) (10%) Find A1 . (c) (10%) Solve AX = b and then write b as a linear combination of the column vectorsin A.
12
A), 3). (a) (6%) Find elementary matrices E1, E2 such that EzElA = I.
(b) (4%) Write A as product of elementary matrices. 4. Let 5. (10%) Suppose A be any n x 72 matrix. Show that A + AT, AAT and ATA are symmetric.
6. (10%) Evaluate the determinants of the following matrices: 2 0, 0 0 A‘ 1 2 B:
1 —2 woor—t 0 1
1 4
6 4
1 1 HOOOOO'! 1 1
0 3
0 0
1 1 7. (10%) Let A and B be 3 x 3 matrices with det(A)= 3 and det(B)= 2.
Evaluate (i) det(ABT), (ii) det(2A‘1). ...
View
Full Document
 Spring '08
 STAFF

Click to edit the document details