3. MATRICES
The word matrix is used in mathematics
to denote a rectangular array of elements in
rows and columns. The elements in the array
are usually numbers, and brackets may be
used to mark the beginning and the end of the
array.
4. MATRIX
An 𝑚 by 𝑛 matrix over a set 𝑆 is a rectangular array of elements of
𝑆, arranged in 𝑚 rows and 𝑛 columns. It is customary to write an 𝑚 by 𝑛
matrix using notation such as
where: A = matrix and 𝑎𝑖𝑗 = element in row i and column j of matrix A.
The matrix 𝐴 is referred to as a matrix of dimension 𝑚 × 𝑛 (read
“𝑚 by 𝑛”).
The 𝑚 × 𝑛 matrix A can be written compactly as 𝐴 = 𝑎𝑖𝑗 𝑚×𝑛
or
simply as 𝐴 = 𝑎𝑖𝑗 if the dimension is known from the context.
5. In compact notation, 𝐵 = 𝑏𝑖𝑗 2×4
is shorthand for the matrix
𝐵 =
𝑏11 𝑏12 𝑏13
𝑏21 𝑏22 𝑏23
𝑏14
𝑏24
As a more concrete example, the matrix A defined by A=
𝑎𝑖𝑗 3×3
with 𝑎𝑖𝑗 = −1 𝑖+𝑗would appear written out as
𝐴 =
1 −1 1
−1 1 −1
1 −1 1
Example 1
8. An 𝑛 × 𝑛 matrix is called a square matrix of order n, and a
square matrix 𝐴 = 𝑎𝑖𝑗 𝑛×𝑛
with 𝑎𝑖𝑗 = 0 whenever 𝑖 ≠ 𝑗 is known as a
diagonal matrix.
Examples:
5 0 0
0 7 0
0 0 −2
and
8 0 0
0 0 0
0 0 8
DIAGONAL MATRIX
9. Matrix EQUALITY
Definition 1.28
Two matrices 𝐴 = 𝑎𝑖𝑗 𝑚×𝑛
and B = 𝑏𝑖𝑗 𝑝×𝑞
over
a set 𝑆 are equal if and only if 𝑚 = 𝑝, 𝑛 = 𝑞, and 𝑎𝑖𝑗 =
𝑏𝑖𝑗 for all pairs 𝑖, 𝑗.
Example:
𝐴 =
1 2 3
4 5 6
𝐵 =
1 2 3
4 5 6
10. Matrix Addition
Definition 1.29
Addition in 𝑀𝑚×𝑛(𝑅) is defined by
𝑎𝑖𝑗 𝑚×𝑛
+ 𝑏𝑖𝑗 𝑚×𝑛
= 𝑐𝑖𝑗 𝑚×𝑛
where 𝑐𝑖𝑗 = 𝑎𝑖𝑗 + 𝑏𝑖𝑗.
To form the sum of two elements in
𝑀𝑚×𝑛(𝑅), we simply add the elements that are placed
in corresponding positions.
12. Matrix Addition
We note that a sum of two matrices with different
dimensions is not defined. For instance, the sum
1 2 0
3 4 0
+
5 6
7 8
is undefined because the dimensions of the two matrices
involved are not equal.
13. Properties of matrix
addition
Addition in 𝑀𝑚×𝑛(𝑅) has the following properties:
a. Addition as defined in Definition 1.29 is a binary
operation on 𝑀𝑚×𝑛(𝑅).
b. Addition is associative in 𝑀𝑚×𝑛(𝑅)
c. 𝑀𝑚×𝑛(𝑅) contains an identity element for addition.
d. Each element of 𝑀𝑚×𝑛(𝑅) has an additive inverse in
𝑀𝑚×𝑛(𝑅).
e. Addition is commutative in 𝑀𝑚×𝑛(𝑅).
14. proof
Let 𝐴 = 𝑎𝑖𝑗 𝑚×𝑛
, 𝐵 = 𝑏𝑖𝑗 𝑚×𝑛
and 𝐶 = 𝑐𝑖𝑗 𝑚×𝑛
be
arbitrary elements of 𝑀𝑚×𝑛(𝑅) .
a. The addition defined in Definition 1.29 is a binary
operation on 𝑴𝒎×𝒏(𝑹)…
because the rule
𝑎𝑖𝑗 + 𝑏𝑖𝑗 = 𝑎𝑖𝑗 + 𝑏𝑖𝑗
yields a result that is both unique and an element of
𝑀𝑚×𝑛(𝑅).
15. proof
b. Addition is associative in 𝑴𝒎×𝒏(𝑹).
The following equalities establish the associative property
for addition.
𝐴 + (𝐵 + 𝐶) = 𝑎𝑖𝑗 + 𝑏𝑖𝑗 + 𝑐𝑖𝑗 by Definition 1.29
= 𝑎𝑖𝑗 + (𝑏𝑖𝑗 + 𝑐𝑖𝑗) by Definition 1.29
= (𝑎𝑖𝑗 + 𝑏𝑖𝑗) + 𝑐𝑖𝑗 since addition in R is associative
= 𝑎𝑖𝑗 + 𝑏𝑖𝑗] + [𝑐𝑖𝑗) by Definition 1.29
= 𝐴 + 𝐵 + 𝐶 by Definition 1.29
17. proof
c. 𝑴𝒎×𝒏(𝑹) contains an identity element for addition.
Let 𝑂𝑚×𝑛 denote the 𝑚 × 𝑛 matrix that has all elements zero. Then
𝐴 + 𝑂𝑚×𝑛= 𝑎𝑖𝑗 𝑚×𝑛
+ 0 𝑚×𝑛
= 𝑎𝑖𝑗 + 0 𝑚×𝑛
by Definition 1.29
= 𝑎𝑖𝑗 𝑚×𝑛
since 0 is the additive identity in R
= 𝐴
A similar computation shows that 𝑂𝑚×𝑛 + 𝐴 = 𝐴, and
therefore 𝑂𝑚×𝑛 is the additive identity for 𝑀𝑚×𝑛(𝑅) , called the
zero matrix of dimension 𝑚 × 𝑛.
18. c. 𝑴𝒎×𝒏(𝑹) contains an identity element for addition.
Let 𝐴 =
1 2
3 4
, then
𝐴 + 0 =
1 2
3 4
+
0 0
0 0
=
1 2
3 4
19. proof
d. Each element of 𝑴𝒎×𝒏(𝑅) has an additive inverse in
𝑴𝒎×𝒏(𝑅).
The matrix –A defined by
−𝐴 = −𝑎𝑖𝑗 𝑚×𝑛
is the additive inverse for A in 𝑀𝑚×𝑛(𝑅).
20. d. Each element of 𝑴𝒎×𝒏(𝑅) has an additive inverse in
𝑴𝒎×𝒏(𝑅).
Let 𝐴 =
1 2
3 4
, then −𝐴 =
−1 −2
−3 −4
21. d. Each element of 𝑴𝒎×𝒏(𝑅) has an additive inverse in
𝑴𝒎×𝒏(𝑅).
This leads to the definition of subtraction in 𝑀𝑚×𝑛(𝑅): For
A and B in 𝑀𝑚×𝑛(𝑅),
𝐴 − 𝐵 = 𝐴 + −𝐵 ,
where −𝐵 = −𝑏𝑖𝑗 is the additive inverse of 𝐵 = 𝑏𝑖𝑗 .
22. d. Each element of 𝑴𝒎×𝒏(𝑅) has an additive inverse in
𝑴𝒎×𝒏(𝑅).
Let 𝐴 =
1 2
3 4
, 𝐵 =
1 3
4 2
Then,
𝐴 + −𝐵 =
1 2
3 4
+
−1 −3
−4 −2
=
0 −1
−1 2
or simply,
𝐴 − 𝐵 =
1 2
3 4
−
1 3
4 2
=
0 −1
−1 2
23. e. Addition is commutative in 𝑴𝒎×𝒏(𝑹).
𝐴 + 𝐵 = 𝑎𝑖𝑗 + 𝑏𝑖𝑗 by Definition 1.29
= 𝑏𝑖𝑗 + 𝑎𝑖𝑗 since addition in R is commutative
= 𝐵 + 𝐴
proof
25. Matrix
multiplication
Definition 1.31
The product of an 𝑚 × 𝑛 matrix 𝐴 over 𝑹 and an 𝑛 × 𝑝 matrix 𝐵 over 𝑹 is an
𝑚 × 𝑝 matrix 𝐶 = 𝐴𝐵, where the element 𝑐𝑖𝑗 in row 𝑖 and column 𝑗 of 𝐴𝐵 is found by
using the elements in row 𝑖 of 𝐴 and the elements in column 𝑗 of 𝐵 in the following
manner:
That is, the element 𝑐𝑖𝑗 = 𝑎𝑖1𝑏1𝑗 + 𝑎𝑖2𝑏2𝑗 + 𝑎𝑖3𝑏3𝑗 + ⋯ + 𝑎𝑖𝑛𝑏𝑛𝑗 in row 𝑖 and
column 𝑗 of 𝐴𝐵 is found by adding the products formed from corresponding elements of
row 𝑖 in 𝐴 and column 𝑗 in 𝐵 (first times first, second times second, and so on).
26. Note that the number of columns in A must equal the
number of rows in B in order to form the product AB. If this is
the case, then A and B are said to be conformable for
multiplication. A simple diagram illustrates this fact.
27. Example 2
Consider the products that can be formed using the matrices.
𝐴 =
1
2
3
and 𝐵 = 4 5 6
𝐴 × 𝐵 =
∎ ∎ ∎
∎ ∎ ∎
∎ ∎ ∎
𝐵 × 𝐴 = ∎
𝐴 × 𝐵 =
1(4) 1(5) 1(6)
2(4) 2(5) 2(6)
3(4) 3(5) 3(6)
𝐵 × 𝐴 = 4 1 + 5 2 + 6(3)
𝐴 × 𝐵 =
4 5 6
8 10 12
12 15 18
𝐵 × 𝐴 = 4 + 10 + 18 = 32
3 x 1 1 x 3
(3 x 3) (1 x 1)
28. Example 3
Consider the products that can be formed using the matrices.
𝐴 =
1 3
2 4
and 𝐵 =
5 7
6 8
𝐴 × 𝐵 =
∎ ∎
∎ ∎
𝐵 × 𝐴 =
∎ ∎
∎ ∎
𝐴 × 𝐵 =
1 5 + 3(6) 1 7 + 3(8)
2 5 + 4(6) 2 7 + 4(8)
𝐵 × 𝐴 =
5 1 + 7(2) 5 3 + 7(4)
6 1 + 8(2) 6 3 + 8(4)
𝐴 × 𝐵 =
23 31
34 46
𝐵 × 𝐴 =
19 29
22 50
2 x 2 2 x 2
(2 x 2) (2 x 2)
29. Example 4
Consider the products that can be formed using the matrices.
Find 𝐴 × 𝐵.
4 x 2 2 x 3
4 x 3
30. Example 4
Consider the products that can be formed using the matrices.
Can 𝐵 × 𝐴 be defined?
NO.
4 x 2 2 x 3
31. Since the number of columns in B is not equal to the
number of rows in A, the product BA is not defined. Similarly,
the products 𝐴 ∗ 𝐴 and 𝐵 ∗ 𝐵 are not defined.
The work in Example 4 shows that multiplication of
matrices does not have the commutative property.
32. Although matrix multiplication is not commutative, it
does have several properties that are analogous to corresponding
properties in the set R of all real numbers.
The sigma notation is useful in writing out proofs of these
properties.
In the sigma notation, the capital Greek letter Σ (sigma) is
used to indicate a sum:
𝑖=1
𝑛
𝑎𝑖 = 𝑎1 + 𝑎2 + 𝑎3 + ⋯ + 𝑎𝑛
33. The variable i is called the index of summation, and the
notations below and above the sigma indicate the value of i at
which the sum starts and the value of i at which it ends. For
example,
𝑖=3
5
𝑏𝑖 = 𝑏3 + 𝑏4 + 𝑏5
34. The index of summation is sometimes called a “dummy
variable” because the value of the sum is unaffected if the index
is changed to a different letter:
𝑖=0
3
𝑎𝑖 =
𝑗=0
3
𝑎𝑗 =
𝑘=0
3
𝑎𝑘 = 𝑎0 + 𝑎1 + 𝑎2 + 𝑎3
35. Using the distributive properties in R, we can write
𝑎
𝑘=1
𝑛
𝑏𝑘 = 𝑎(𝑏1 + 𝑏2 + 𝑏3 + ⋯ + 𝑏𝑛)
= 𝑎𝑏1 + 𝑎𝑏2 + 𝑎𝑏3 + ⋯ + 𝑎𝑏𝑛
=
𝑘=1
𝑛
𝑎𝑏𝑘
Similarly,
𝑘=1
𝑛
𝑏𝑘 𝑎 =
𝑘=1
𝑛
𝑏𝑘𝑎
36. In the definition of the matrix product AB, the element
𝑐𝑖𝑗 = 𝑎𝑖1𝑏1𝑗 + 𝑎𝑖2𝑏2𝑗 + ⋯ + 𝑎𝑖𝑛𝑏𝑛𝑗
can be written compactly by use of the sigma notation as
𝑐𝑖𝑗 =
𝑘=1
𝑛
𝑎𝑖𝑘𝑏𝑘𝑗
37. Properties of matrix
multiplication
1. Associative Property of Multiplication
2. Distributive Properties
3. Special Properties of In
4. Left Identity, Right Identity
38. Properties of matrix
multiplication
Associative Property of Multiplication
Let 𝐴 = 𝑎𝑖𝑗 𝑚×𝑛
, 𝐵 = 𝑏𝑖𝑗 𝑛×𝑝
and
𝐶 = 𝑐𝑖𝑗 𝑝×𝑞
be matrices over R. Then 𝐴(𝐵𝐶) =
(𝐴𝐵)𝐶.
Proof:
𝐵𝐶 = 𝑑𝑖𝑗 𝑛×𝑞
𝑑𝑖𝑗 =
𝑘=1
𝑛
𝑏𝑖𝑘𝑐𝑘𝑗
𝐴(𝐵𝐶) =
𝑟=1
𝑛
𝑎𝑖𝑟𝑑𝑟𝑗
𝑚×𝑞
𝑟=1
𝑛
𝑎𝑖𝑟𝑑𝑟𝑗 =
𝑟=1
𝑛
𝑎𝑖𝑟
𝑘=1
𝑝
𝑏𝑟𝑘𝑐𝑘𝑗
=
𝑟=1
𝑛
𝑘=1
𝑝
𝑎𝑖𝑟(𝑏𝑟𝑘𝑐𝑘𝑗)
40. The last equality follows from the associative property
𝑎𝑖𝑟(𝑏𝑟𝑘𝑐𝑘𝑗) = (𝑎𝑖𝑟𝑏𝑟𝑘)𝑐𝑘𝑗
of multiplication of real numbers. Comparing the elements in row i,
column j, of A(BC)
and (AB)C, we see that
𝑟=1
𝑛
𝑘=1
𝑝
𝑎𝑖𝑟(𝑏𝑟𝑘𝑐𝑘𝑗) =
𝑟=1
𝑛
𝑘=1
𝑝
𝑎𝑖𝑟(𝑏𝑟𝑘𝑐𝑘𝑗)
∴ 𝐴 𝐵𝐶 = 𝐴𝐵 𝐶
42. Properties of matrix
multiplication
Distributive Properties of Multiplication
Let A be an 𝑚 × 𝑛 matrix over R, let B and C be 𝑛 × 𝑝
matrices over R, and let D be a 𝑝 × 𝑞 matrix over R. Then,
a. 𝐴(𝐵 + 𝐶) = 𝐴𝐵 + 𝐴𝐶, and
b. (𝐵 + 𝐶)𝐷 = 𝐵𝐷 + 𝐶𝐷
45. For each positive integer 𝑛, we define a special matrix 𝐼𝑛 by
𝐼𝑛 = 𝛿𝑖𝑗 𝑛×𝑛
where 𝛿𝑖𝑗 =
1 if 𝑖 = 𝑗
0 if 𝑖 ≠ 𝑗
(The symbol 𝛿𝑖𝑗 used in defining 𝐼𝑛 is called the Kronecker delta.) For
𝑛 = 2 and 𝑛 = 3, these special matrices are given by
𝐼2 =
1 0
0 1
and 𝐼3 =
1 0 0
0 1 0
0 0 1
The matrices 𝐼𝑛 have special properties in matrix multiplication.
46. Properties of matrix
multiplication
Special Properties of 𝑰𝒏
Let 𝐴 be an arbitrary 𝑚 × 𝑛 matrix over R. With 𝐼𝑛 as
defined in the preceding paragraph,
a. 𝐼𝑚𝐴 = 𝐴, and
b. 𝐴𝐼𝑛 = 𝐴
49. Group 1
1. Write out the matrix that matches the
given description.
𝐴 = 𝑎𝑖𝑗 3×2
with 𝑎𝑖𝑗 = 2𝑖 − 𝑗
2. Add.
3. Multiply.
50. Group 2
1. Write out the matrix that matches the
given description.
𝐴 = 𝑎𝑖𝑗 2×3
with 𝑎𝑖𝑗 = 1 if 𝑖 < 𝑗 and
𝑎𝑖𝑗 = 0 if 𝑖 ≥ 𝑗.
2. Add.
−1 3 −2
3 −1 5
+
1 2 −3
6 2 −5
3. Multiply.
51. Group 3
1. Write out the matrix that matches
the given description.
𝐴 = 𝑎𝑖𝑗 2×3
with 𝑎𝑖𝑗 = −1 if 𝑖 = 𝑗
and 𝑎𝑖𝑗 = 0 if 𝑖 ≠ 𝑗.
2. Add.
2 −4 4
−3 0 2
+
−1 −2 4
−5 2 −1
3. Multiply.
