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Matrices
Prerequisites: Adding, subtracting, multiplying and dividing numbers;
elementary row operations.
Maths Applications: Solving systems of equations; describing geometric
transformations; deriving addition formulae.
Real-World Applications: Balancing chemical equations; flight
stopover information; currents in electrical
circuits; formulation of fundamental physical
laws.
Basic Definitions
Definition:
A matrix is a rectangular array of numbers (aka entries or elements) in
parentheses, each entry being in a particular row and column.
Definition:
The order of a matrix is given as m × n (read m by n), where m is the
number of rows and n the number of columns and is written as,
A ≡ (
ij
a )
m n×
def
11 12 13 1
21 22 23 2
31 32 33 3
1 2 3
n
n
n
m m m mn
a a a a
a a a a
a a a a
a a a a
The element in row i and column j of a matrix is written as
ij
a and called
the (i, j)
th
entry of A.
In this course, we will deal almost exclusively with matrices that have
orders 2 × 2 and 3 × 3.
Definition:
The main diagonal (aka leading diagonal) of any matrix is the set of
entries
ij
a where i = j.
Special cases arise when either m = 1 or n = 1.
Definition:
A row matrix is a 1 × n matrix and is written as,
( )
n n
a a a a
11 12 1( −1) 1
A column matrix is a m × 1 matrix and is written as,
m
m
a
a
a
a
11
21
( 1)
1
−
The case when m = n is a very important one.
Definition:
A square matrix (of order m × ××
× m) is a matrix with the same number of
rows as columns (equal to m) and is written as,
11 12 1
21 22 2
1 2
m
m
m m mm
a a a
a a a
a a a
Definition:
The identity matrix (of order m) is the m × m matrix all of whose
entries are 0 apart from those on the main diagonal, where they all equal
Example 1
Add the matrices A =
and B =
As the matrices have the same order, they can be added.
A + B =
Example 2
Find the difference P − Q where P =
and
Q =
P − Q =
Definition:
The scalar multiplication of A by k ( k ∈ ) is obtained by multiplying
each entry of A by k,
ij
ka
def
ij
ka
Example 3
If A =
, calculate
1
2
A.
1
2
A =
1
2
5 1
2 2
9 1
2 2
Matrix Multiplication
Definition:
The matrix product of A and B, where A is of order m × n and B is of
order n
×
p is obtained by the following prescription,
ij
ab
def
1
n
ik kj
k
a b
=
∑
( 1 ≤ i ≤ mand 1 ≤ j ≤ p)
Note that the number of columns of A must equal the number of rows of
B. To see how to use the horrible prescription, split up A into rows and B
into columns; then taking the ‘ scalar product ’ of the i
th
row of A with
the j
th
column of B gives the ( i, j )
th
entry of AB.
BA =
BA =
Definition:
When forming AB, B is pre-multiplied by A and A is post-multiplied by
B.
Definition:
The transpose of A (with the order of A being m × n), denoted by
T
A
(sometimes A′ ), is the n × m matrix obtained by interchanging the rows
and columns of A,
T
(a )
ij
def
ji
a
Example 6
Find the transpose of P =
T
P =
Definition:
A (square) matrix A can be multiplied by itself any number of times,
giving the n
th
power of A,
n
A
def
n times
A × A × A × … × A
Basic Properties of Matrices
- A + B = B + A
- ( A + B ) + C = A + ( B + C )
- k ( A + B ) = kA + kB
- ( A + B )
T
T
A +
T
B
T T
( A ) = A
T
( kA ) =
T
kA
- A ( BC ) = ( AB ) C
- A ( B + C ) = AB + AC
T
( AB ) =
T T
B A
m
A
n
A =
m n
A
n
A
m
A
There are 3 important properties that are worth singling out separately.
- A + O = O + A = A
- A I = A I = A
- A O = A O = O
Thus, the identity and zero matrices behave like the numbers 1 and 0
respectively in ordinary arithmetic and algebra.
Example 7
For the matrix A =
, show that
2
A
= pA +
2
qI
, stating the
values of the integers p and q. Hence write
3
A in the form gA +
2
hI ,
stating the values of g and h.
Example 8
Show that
T
( kABC ) = k
T T T
C B A.
T
( kABC ) = k
T
( ABC)
= k
T
(( AB C) )
= k
T
C
T
( AB)
= k
T
C
T T
( B A )
= k
T
C
T T
B A
Make sure you can justify each equality in Example 8.
One matrix property that has no counterpart in ordinary arithmetic and
algebra is the fact that the product of 2 matrices can be zero without
either of the matrices being the zero matrix.
Example 9
Calculate AB for the matrices A =
and B =
AB =
Special Types of Matrices
Symmetric and Skew-Symmetric Matrices
Definition:
A matrix A is symmetric if,
T
A
= A
Note that a symmetric matrix must be square.
Example 10
Show that the matrix A =
is symmetric.
T
A =
T
Interchanging the rows and columns of A gives,
T
A =
T
A = A
Hence, as
T
A = A, A is symmetric.
Definition:
A matrix A is skew-symmetric (aka anti-symmetric) if,
T
A = − A
2
I
Hence, as
T
R R equals the identity matrix, R is orthogonal.
Determinants
The solution to the 1 × 1 system,
ax = b
is,
x =
b
a
assuming a ≠ 0.
The solution to the 2 ×
2 system of equations,
ax by e
cx dy f
is,
x =
de bf
ad bc
y =
af ce
ad bc
assuming ad − bc ≠ 0.
The solution to the 3 × 3 system of equations,
ax by cz j
dx ey fz k
gx hy iz l
is (this takes a lot more effort),
x =
j ei fh k bi ch l bf ce
a ei fh b di fg c dh eg
y =
j fg di k cg ai l cd af
a ei fh b di fg c dh eg
z =
j dh ge k ah bg l ae bd
a ei fh b di fg c dh eg
assuming a ei( − fh ) − b di( − fg ) + c dh( − eg) ≠ 0.
In each of these solutions, we require the denominators to be non-zero.
The denominators that arise in the solutions have a pattern (not
necessarily that obvious !) and a special name. We first need some
definitions.
Definition:
A permutation, denoted by σ , of an ordered set of numbers (1, 2, 3, … ,
n) is a rearrangement of those numbers.
An even permutation is one where the rearrangement involves an even
number of consecutive switches starting from the original numbers.
An odd permutation is one where the rearrangement involves an odd
number of consecutive switches starting from the original numbers.
Definition:
The sign of a permutation σ , denoted by sign σ , is defined to be + 1
for an even permutation and − 1 for an odd permutation.
Example 13
For the case n = 3, the permutation (1, 2, 3)
σ
→ (2, 1, 3) is odd, as
the result of σ involves only 1 switch (1 ↔ 2) between consecutive
numbers of (1, 2, 3). The sign of this permutation is − 1.
Theorem:
The determinant of an n × n matrix is given by the Laplace expansion
formula,
det ( A ) =
1
n
ij ij
j
a C
=
∑
( i = 1, 2, 3, … , n)
The Laplace expansion formula expresses the determinant of a matrix in
terms of smaller determinants. For satisfaction and reassurance, the
following theorems should be proven using the Laplace expansion formula.
Theorem:
The determinant of a 1 × 1 matrix is,
A ≡ ( a ) = a
Theorem:
The determinant of a 2 ×
2 matrix is,
A ≡
a b
c d
= ad − bc
Theorem:
The determinant of a 3 × 3 matrix is,
A ≡
a b c
d e f
g h i
= a
e f
h i
− b
d f
g i
+ c
d e
g h
Notice that these are precisely the expressions for the denominators for
the systems at the start of this section.
Example 15
Calculate the determinant of the matrix B =
det ( B ) =
Example 16
Calculate the determinant of the matrix F =
det ( F ) =
Example 17
Solve the equation
3 x 5
= 21 for x.
3 x 5
10 + 12 x = 21
Definition:
The cofactor matrix of a square matrix A is the matrix C whose ( i, j )
th
entry is
ij
C.
Definition:
The adjugate (aka classical adjoint) of a square matrix A is the
transpose of the cofactor matrix C,
adj ( A )
def
T
C
Theorem:
The inverse of a matrix A is given by,
1
A
−
adj ( )
det ( )
A
A
Theorem:
A matrix is invertible iff det ( A ) ≠ 0.
This theorem implies that a matrix is non-invertible iff det ( A ) = 0.
Example 18
Show that A =
is invertible.
det ( A ) =
Hence, as det ( A ) ≠ 0, A is invertible.
Example 19
Determine the values of k for which the matrix
k
k
is singular.
For singularity, we require the determinant of the given matrix to be 0.
k
k
k ( k + 5) − ( −2). ( −2) = 0
2
k + 5 k + 4 = 0
( k + 1) ( k + 4) = 0
Hence, the given matrix is singular when k = −1 and k = −4.
Example 20
If the matrix A satisfies the equation
2
A = 18 A − 27
2
I , show
(without explicitly calculating
1
A
−
) that
1
A
−
= DA +
2
EI
, stating the
values of D and E.
2
A = 18 A − 27
2
I
Multiplying (doesn’t matter whether post or pre, as the only matrices
involved are A and
2
I ) this equation throughout by
1
A
−
gives,
1
A
− 2
A =
1
A
−
(18 A − 27
2
I )
Performing the multiplications and simplifying gives,
A = 18
2
I − 27
1
A
−
