Practice Problems for Exam 1
1. Given the matrix A and vector ~b below, using only rational numbers,
find the following: (a) RREF of [A|~b] (the intermediate elimination
steps need not be defined); (b) the number of free variables of A and
what they are; (c) homogeneous solution of A~xH = ~0 with ~xH written
in vector form; and (d) the general solution of A~x = ~b written in vector
form.




A=


3
0
5 −2
6
−9
1 −17
10 −18 


15 14 −3
49
25 
6 −3
16 −13
7

~b = 


−29
148
725
−225





2. Given A = LU where L and U are defined below along with ~b, perform
forward and backward substitution to solve A~x = ~b. Show intermediate steps.





1 0 0
−3 7 2
1





A~x =  −4 1 0   0 3 2  ~x = ~b =  7 
−4 5 1
0 0 5
1
3. (a) Find the LU decomposition of A. The diagonal position of L
must be 1. (b)Using information from the LU decomposition, find the
det(A).


2 −3 4


7 3 
A= 1
5
2 2
4. Without using a calculator, find the determinant of A where




A=
−2
7
1
3
3
1
1
0
0
4
2
3
0
1
0 −5





5. Given ~v = [2, −1, 8]T and w
~ = [3, 2, 0]T , answer the following
(a) What is the dimension of ~v T w?
~
(b) What is the dimension of ~v w
~T ?
(c) Is the angle between ~v and w
~ obtuse or actue? Provide a reason
for the answer.
(d) Find proj ~v w.
~
(e) Find proj w~ ~v .
(f) Find proj~v ~v . Does this make sense?
6. Find the general formula for A−1 given A below. When is the matrix
not invertible?


a b 0


A= c d 0 
0 0 e
. Hint: Consider blocking A.
7. If det(A) = 7, det(B) = −3, det(C) = 4, det(DT ) = −2, and det(E) =
0, what are the following values (if they exist)? Assume the matrices
are all square and of the same dimention.
(a) det(A−1 C T D2 )
(b) det(A50 E)
(c) det(B T D−1 )
8. Given ~v = [a, −2, 3]T , w
~ = [4, 1, 7] and ~u = [5, b, 3] answer the following.
(a) Find the values of a such that ||~v || = 5
(b) Find the values of a such that ||~v || = 3
(c) Find a such that ~v and w
~ are orthongonal.
(d) When are ~v and ~u orthogonal?
(e) Is it possible to find values of a and b such that ||~v || = ||~u||? Why
or why not?