Mathematics 133 Summer 2010,
MIDTERM EXAMINATION, May 17, 5:00 - 6:30 pm
Exam consists of 4 written questions of unequal weight, 25p in total. Show and justify each
step in the solutions of these question. Answers have be written at the back of this page. All
paper with work must be attached.
1. (6p.) Let A=
0 1 0 0 −2
1 5 −3 0 −10
2 4 −6 1 −3
3 19 −9 0 −38
,C=
−3
−14
−8
−54
.
(a) Find a matrix Bso that B A is the reduced row echelon form of A.
(b) Find a general solution to AX =C, write it in a vector-parametric form.
(c) Let Ribe the i’th row of A. There are (4 −rk A) linear dependence relations
among Ri’s, find them.
2. (5p.) Let A=3 0
−2
34.
(a) Compute the inverse of Ausing the book-keeping matrix.
(b) Write Aas a product of three elementary matrices.
3. (7p.) Let kbe a real number and let A=
−1−2−1
3 9 3k
2k+ 5 4
.
(a) Compute det Ausing ONLY the knowledge about the behavior of det under ele-
mentary row operations and the fact that det I3= 1. Hint: [0,0, x] = x[0,0,1].
(b) Compute det Ausing the cofactor expansion along the second column.
(c) For which kdoes the system AX = [1,√2,−3]Thave a unique solution? Justify
your answer.
(d) For k=−1 compute the (2,1)-entry of the adjoint matrix of A.
4. Let P= [2,−2,1]T,Q= [0,0,1]T,R= [0,2,0]T,S= [2,−3,1]T. And let π=
Q+ Span{R, S}.
(a) (1p.) Does the point [0,3,5]Tbelong to the plane π?
(b) (6p.) Compute the distance between Pand π. If some formulas for the distance
are used they must be proved.
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