234 Higher Engineering Mathematics
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
[(3 × 2)
[(3 × (−5))
+ (4 × 5)
+(4 × (−6))
+ (0 × (−1))]
+(0 × (−7))]
[(−2 × 2)
[(−2 × (−5))
+ (6 × 5)
+(6 × (−6))
+ (−3 × (−1))]
+(−3 × (−7))]
[(7 × 2)
[(7 × (−5))
+ (−4 × 5)
+(−4 × (−6))
+ (1 × (−1))]
+(1 × (−7))]
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎝
26 −39
29 −5
−7 −18
⎞
⎠
Problem 8. Determine
⎛
⎝
1 0 3
2 1 2
1 3 1
⎞
⎠ ×
⎛
⎝
2 2 0
1 3 2
3 2 0
⎞
⎠
The sum of the products of the elements of each row of
the first matrix and the elements of each column of the
second matrix are taken one at a time. Thus:
⎛
⎝
1 0 3
2 1 2
1 3 1
⎞
⎠ ×
⎛
⎝
2 2 0
1 3 2
3 2 0
⎞
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
[(1 × 2)
[(1 × 2)
[(1 × 0)
+ (0 × 1)
+ (0 × 3)
+ (0 × 2)
+ (3 × 3)] + (3 × 2)] + (3 × 0)]
[(2 × 2)
[(2 × 2)
[(2 × 0)
+ (1 × 1)
+ (1 × 3)
+ (1 × 2)
+ (2 × 3)] + (2 × 2)] + (2 × 0)]
[(1 × 2)
[(1 × 2)
[(1 × 0)
+ (3 × 1)
+ (3 × 3)
+ (3 × 2)
+ (1 × 3)] + (1 × 2)] + (1 × 0)]
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎝
11 8 0
11 11 2
8 13 6
⎞
⎠
In algebra, the commutative law of multiplication states
that a × b = b × a. For matrices, this law is only true in
a few special cases, and in general A × B is not equal
to B × A.
Problem 9. If A =
2 3
1 0
and
B =
2 3
0 1
show that A × B = B × A.
A × B =
2 3
1 0
×
2 3
0 1
=
[(2 × 2) + (3 × 0)] [(2 × 3) + (3 × 1)]
[(1 × 2) + (0 × 0)] [(1 × 3) + (0 × 1)]
=
4 9
2 3
B × A =
2 3
0 1
×
2 3
1 0
=
[(2 × 2) + (3 × 1)] [(2 × 3) + (3 × 0)]
[(0 × 2) + (1 × 1)] [(0 × 3) + (1 × 0)]
=
7 6
1 0
Since
4 9
2 3
=
7 6
1 0
, then A × B = B × A
Now try the following exercise
Exercise 93 Further problems on addition,
subtraction and multiplication of matrices
In Problems 1 to 13, the matrices A to K are:
A =
3 −1
−4
7
B =
5 2
−1 6
C =
−1.3
7.4
2.5 −3.9
D =
⎛
⎝
4 −7
6
−2
4
0
5
7 −4
⎞
⎠
E =
⎛
⎝
3
6 2
5 −3 7
−1
0 2
⎞
⎠
F =
⎛
⎝
3.1 2.4
6.4
−1.6 3.8 −1.9
5.3 3.4 −4.8
⎞
⎠ G =
6
−2
H =
−2
5
J =
⎛
⎝
4
−11
7
⎞
⎠ K =
⎛
⎝
1 0
0 1
1 0
⎞
⎠
Addition, subtraction and multiplication
In Problems 1 to 12, perform the matrix operation
stated.
1. A + B
8 1
−5 13
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
[(3 × 2)
[(3 × (−5))
+ (4 × 5)
+(4 × (−6))
+ (0 × (−1))]
+(0 × (−7))]
[(−2 × 2)
[(−2 × (−5))
+ (6 × 5)
+(6 × (−6))
+ (−3 × (−1))]
+(−3 × (−7))]
[(7 × 2)
[(7 × (−5))
+ (−4 × 5)
+(−4 × (−6))
+ (1 × (−1))]
+(1 × (−7))]
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎝
26 −39
29 −5
−7 −18
⎞
⎠
Problem 8. Determine
⎛
⎝
1 0 3
2 1 2
1 3 1
⎞
⎠ ×
⎛
⎝
2 2 0
1 3 2
3 2 0
⎞
⎠
The sum of the products of the elements of each row of
the first matrix and the elements of each column of the
second matrix are taken one at a time. Thus:
⎛
⎝
1 0 3
2 1 2
1 3 1
⎞
⎠ ×
⎛
⎝
2 2 0
1 3 2
3 2 0
⎞
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
[(1 × 2)
[(1 × 2)
[(1 × 0)
+ (0 × 1)
+ (0 × 3)
+ (0 × 2)
+ (3 × 3)] + (3 × 2)] + (3 × 0)]
[(2 × 2)
[(2 × 2)
[(2 × 0)
+ (1 × 1)
+ (1 × 3)
+ (1 × 2)
+ (2 × 3)] + (2 × 2)] + (2 × 0)]
[(1 × 2)
[(1 × 2)
[(1 × 0)
+ (3 × 1)
+ (3 × 3)
+ (3 × 2)
+ (1 × 3)] + (1 × 2)] + (1 × 0)]
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎝
11 8 0
11 11 2
8 13 6
⎞
⎠
In algebra, the commutative law of multiplication states
that a × b = b × a. For matrices, this law is only true in
a few special cases, and in general A × B is not equal
to B × A.
Problem 9. If A =
2 3
1 0
and
B =
2 3
0 1
show that A × B = B × A.
A × B =
2 3
1 0
×
2 3
0 1
=
[(2 × 2) + (3 × 0)] [(2 × 3) + (3 × 1)]
[(1 × 2) + (0 × 0)] [(1 × 3) + (0 × 1)]
=
4 9
2 3
B × A =
2 3
0 1
×
2 3
1 0
=
[(2 × 2) + (3 × 1)] [(2 × 3) + (3 × 0)]
[(0 × 2) + (1 × 1)] [(0 × 3) + (1 × 0)]
=
7 6
1 0
Since
4 9
2 3
=
7 6
1 0
, then A × B = B × A
Now try the following exercise
Exercise 93 Further problems on addition,
subtraction and multiplication of matrices
In Problems 1 to 13, the matrices A to K are:
A =
3 −1
−4
7
B =
5 2
−1 6
C =
−1.3
7.4
2.5 −3.9
D =
⎛
⎝
4 −7
6
−2
4
0
5
7 −4
⎞
⎠
E =
⎛
⎝
3
6 2
5 −3 7
−1
0 2
⎞
⎠
F =
⎛
⎝
3.1 2.4
6.4
−1.6 3.8 −1.9
5.3 3.4 −4.8
⎞
⎠ G =
6
−2
H =
−2
5
J =
⎛
⎝
4
−11
7
⎞
⎠ K =
⎛
⎝
1 0
0 1
1 0
⎞
⎠
Addition, subtraction and multiplication
In Problems 1 to 12, perform the matrix operation
stated.
1. A + B
8 1
−5 13