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
⎛
⎝
103
212
131
⎞
⎠×
⎛
⎝
220
132
320
⎞
⎠
The sum of the products of the elements ofeach row of
the first matrix and the elements of each column of the
second matrix are taken one at a time. Thus:
⎛
⎝
103
212
131
⎞
⎠×
⎛
⎝
220
132
320
⎞
⎠
=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
[( 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
8136
⎞
⎠
In algebra, the commutative law of multiplicationstates
thata×b=b×a. For matrices, this law is only true in
a few special cases, and in generalA×Bisnotequal
toB×A.
Problem 9. IfA=
(
23
10
)
and
B=
(
23
01
)
show thatA×B=B×A.
A×B=
(
23
10
)
×
(
23
01
)
=
(
[( 2 × 2 )+( 3 × 0 )][( 2 × 3 )+( 3 × 1 )]
[( 1 × 2 )+( 0 × 0 )][( 1 × 3 )+( 0 × 1 )]
)
=
(
49
23
)
B×A=
(
23
01
)
×
(
23
10
)
=
(
[( 2 × 2 )+( 3 × 1 )][( 2 × 3 )+( 3 × 0 )]
[( 0 × 2 )+( 1 × 1 )][( 0 × 3 )+( 1 × 0 )]
)
=
(
76
10
)
Since
(
49
23
)
=
(
76
10
)
,thenA×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 matricesAtoKare:
A=
(
3 − 1
− 47
)
B=
(
52
− 16
)
C=
(
− 1. 37. 4
2. 5 − 3. 9
)
D=
⎛
⎝
4 − 76
− 240
57 − 4
⎞
⎠
E=
⎛
⎝
362
5 − 37
− 102
⎞
⎠
F=
⎛
⎝
3. 12. 46. 4
− 1. 63. 8 − 1. 9
5. 33. 4 − 4. 8
⎞
⎠ G=
(
6
− 2
)
H=
(
− 2
5
)
J=
⎛
⎝
4
− 11
7
⎞
⎠ K=
⎛
⎝
10
01
10
⎞
⎠
Addition, subtraction and multiplication
In Problems 1 to 12, perform the matrix operation
stated.
- A+B
[(
81
− 513
)]