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Unformatted text preview: Math 310: Hour Exam 1 (Solutions)
Prof. S. Smith: Mon 28 Sept 1998 Problem 1: (a) Using Gauss—Jordan, ﬁnd the row—reduced echelon form of the following aug—
mented matrix. (INDICATE the row operations you use). A?“ 1 2 3 6 AfgxgvAéxg 1 0 —1 0
—> 0 1 2 3 —> 0 1 2 3
0 —1 —2 —3 0 0 0 0 (b) Then give the solutions of the corresponding linear system A50 : I).
So solutions are (7‘, 3 — 27‘, 7‘) for free variable x3 = 7‘. Problem 2: (a) What elementary row OPERATION Will change 12 1 2
_ _ 7
A (2 1) toB (0 _3). A?“ : add —2 times 1st row to 2nd.
(b) What elementary row MATRIX E will, by left muliplication, perform the same operation? (that is, EA = B) 1 . . . . . .
E = < _2 (1) > : (obtained by domg that operation to the 1dent1ty matrix). Problem 3: (a) Find the inverse (any method) of: 1 0 0
A : ( 1 1 0 ) .
I I 1
Using (All) method: use A§1X1,A§1X1 to clear first column; then A§1X2 to clear second column.
1 0 0
Get inverse: —1 1 0
0 —1 0 (b) Give the LU —decomposition of 12
A—(m) that is, ﬁnd lowertriangular L and uppertriangular U, so that A = LU. Apply 142—1“ to get U : < (1) i 80 L from inverse operation A31“ is ( 1 [1) > Problem 4: (a) Find the determinant of Top row: 1(1.1 — 0.1) — 1(0.1 — 1.1) + 0 = (1) — (—1) = 2.
(b) Use Cramer’s rule to solve
1 2 4
3 4 1 10 det(A) = 1.4 — 3.2 = —2, so 1161 = —§(4.4 — 10.2): 2 and $2 = — Her—1
Ol—‘H
1—11—10 (1.10 — 3.4) = 1. 1
2 Problem 5: (a) Is (1,0,1) in the span of (1,1,1) and (1,2,1)?
Either give coefﬁcients in a linear combination, or explain Why it is not possible. 1 1 1
ROWreduction quickly produces coeﬂicients 2 and —1. (b) Let V be the space of 2 X 2 matrices, and W the subSET of diagonal matrices.
Show that W is a subSPACE of V. A E W says diagonal, meaning 0 of the diagonal, so A has form < a 0 1 1 1
Yes: Set up augmented matrix (Ab) : ( 1 2 0 ) . 0 b 0 d a+c 0
0 b+d Similarly ifB E W it has form < C 0 > Is A + B E W? It is < ), also diagonal, so yes. m 0
0 Tb
We see “yes”, W is a subspace of V. For scalar 7‘, is TA E W? It is < >, also diagonal, so yes. ...
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 Fall '08
 Smith

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