MAT 0AA2
18/02/2015
Tutorial 2 [10]
I.
Given two matrices
[5]
A
=
3
2

2
1
5
3

1
0
1
B
=

1
3
3
0

1
2
2
1
4
Use the row method or the column method to
find
1. the second column of
AB
2. the third row of
AB
3. Express the matrix
E
below as a linear com
bination of the matrices
A
and
B
E
=
9

5

13
2
13
0

8

3

10
1
II.
Given two matrices
[5]
M
=
1
1
0
1
0
2
0
2

3
and
R
=
h
k
1
1
i
1. Find all values of
k
, if any, that satisfy the
equation
RMR
T
= 0
2. Express the matrix
C
=
0
1

1
as a linear
combination of columns of
M
3. Compare Tr(
M
) and Tr(
M
T
)
Section 5.1: Eigenvalues and eigen
vectors
Examples
Example
Compute the following matrix multiplications:
1.
"
3
1
1
3
# "
1
1
#
2.
"
1
0
0

1
# "
0
1
#
3.
"
1
0
0

1
# "
1
0
#
4.
8

9
4
3

4
3

3
3
1
1
2
3
Do you notice anything interesting?
2
Definitions
Definition
If
A
is an
n
×
n
matrix, then a nonzero vector
x
in
R
n
is called an
eigenvector
of
A
if
A
x
is a scalar
multiple of
x
, i.e.,
A
x
=
λ
x
for some scalar
λ
. The scalar
λ
is called an
eigen
value
of
A
and
x
is said to be an
eigenvector cor
responding to
λ
.