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MA 405H Test 1 - MA405H TEST 1 Department of Mathematics...

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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 A-1 . (c) (10%) Solve AX = b and then write b as a linear combination of the column vectors-in 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). ...
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