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)
andB=(
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×ANow try the following exerciseExercise 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)]