Name:
CSU ID:
Homework 4
February 10, 2016
1. Find the determinant of A. Specify whether the matrix has an inverse
without trying to compute the inverse.
"
A=
2
5
1 −7
#
2. Apply the directions in the previous problem to




A=
2
8
4
0

1 −5 −15
0
3
1 


6
1
11 
4
3
17
3. Refer to A appearing in the previous problem. Find (a) det[4A−1 ] and
(b) det[(4A)−1 ]
4. (a) Find the elementary matrices needed to reduce A to the identity.
An example of an elementary matrix is one such that the values
in the (2,1) and (3,1) are eliminated at the same time. Follow as
it is discussed in class.


2 −6 1


A =  12 −33 11 
−6
30 15
(b) Using the elementary matrices from part (a), find the det(A).
Show how the conclusion was reached.
(c) Using the LU decomposition, find the determinant. Is part(c)
easier than part(b)?
5. Given the LU decomposition of A is as follows, what is det(A)?



1 0 0
−7 0 2



A = LU =  −5 1 0   0 4 8 
7 3 1
0 0 3
6. Find a formula for the inverse of the matrix


e 0 f


A =  0 a 0 .
c 0 d
What restrictions on a, b, c, d, e are needed to guarantee that an inverse
exists?
7. Given A, B, C and D are all square matrices of the same dimension
with det(A) = 5, det(B) = −3, det(C) = 1/2, and det(D) = −3 find
(a) det(A−1 B T C)
(b) det((BA)T C 2 )
(c) det(D3 B −1 AC)
8. Given ~x = [−3, 2, 5]T and A as in three problems back, find ||~x|| and
||A~x||.
9. S3.1 ]6
10. S3.1 ]16
11. S3.1 ]20
12. S3.1 ]TF
13. S3.2 (a)]5,(b)]6
14. S3.2 ]22
15. S3.2 ](a)7,(b)8
16. S3.2 Rewrite v = [2, −3, 3]T as sum of the projections of v onto w =
[−3, 4, 1] and a vector that is orthongal to w.
17. DO THIS BEFORE THE EXAM
(a) Find an equation of a plane containing the points (−2, 0, 5), (4, 1, 1), (3, −8, 2).
(b) Find the area of the triangle whose vertices are the points in part
(a).
(c) Find the distance from the point (-1,0,5) to the plane.
18. DO THIS BEFORE THE EXAM S3.3 ]18
19. DO THIS BEFORE THE EXAM S3.3 ]26
20. DO THIS BEFORE THE EXAM S3.3 ]28