Review Exam
1.
a) Let
 
1
2
 
x =  . .
 .. 
10
Find the entry in the second row and the fifth column of xxT .
1
2.
a) Let
1
A=
5
2
6
3
7
4
.
8
If you think of A as a function taking vectors to vectors, what are
the dimensions of the input vectors and the output vectors.
b) Give 5 different vectors b such that the equation
Ax = b
has a solution x.
2
3.
a) Let A3×4 be a matrix given as

a11 a12
A = a21 a22
a31 a32
a13
a23
a33

a14
a24  .
a34
 
3
For b = 0, let the set of solutions to Ax = b be given as
1
n
x1
x2
x3
x4
T
o
: x1 + 3x2 + 4x3 − x4 = 1 .
 
a11
What is the first column of A, i.e. find a = a21 .
a31
b) What are the other columns of A?
3
4.
Find the set of solutions to the system of linear equations represented
by the augmented matrix


1 2 3 4 1
 2 3 4 5 1 
3 4 5 6 1
4
5.
a) Write a 3 × 5 matrix in reduced row echelon form such that the
corresponding linear system of equations has two lead variables
and three free variables.
b) Let A be the matrix you wrote in part a). Give a single solution
 
x1
 
x2 
2
 


x
3 .
x=
satisfying
Ax
=
 3
x4 
0
x5
5
6.
Let

a11
A = a21
a31
a12
a22
a32

a13
a23  ,
a33
such that
 
x1
Solve the system Ax = b for x = x2  , and
x3
 
 
1
0
a) b = 0
b) b = 1
0
0
6
A−1

1
= 1
1
1
2
2

1
3 .
4
 
1
c) b = 1
1
7.
Decompose (factor) the following
matrices.

0
0
2
7
matrix into a product of elementary

0 1
3 2
6 0
8.
Find the determinant of the following matrices.


1 2 3
a) A = 0 2 3
0 0 3

0
0
b) B = 
0
1
0
1
0
0
1
0
0
0

0
0

1
0

1 2 42
3 4 34
c) For C = 
0 0 5
0 0 7

86
−9
 see that det(C) = (1·4−2·3)(5·8−7·6).
6
8
8