Higher Engineering Mathematics, Sixth Edition

The theory of matrices and determinants 233

For scalar multiplication,each element is multiplied by
the scalar quantity, hence

2 A= 2

(

− 30

7 − 4

)

=

(

− 60

14 − 8

)

3 B= 3

(

2 − 1

− 74

)

=

(

6 − 3

−21 12

)

and 4C= 4

(

10

− 2 − 4

)

=

(

40

− 8 − 16

)

Hence 2A− 3 B+ 4 C

=

(

− 60

14 − 8

)

−

(

6 − 3

−21 12

)

+

(

40

− 8 − 16

)

=

(

− 6 − 6 + 40 −(− 3 )+ 0

14 −(− 21 )+(− 8 ) − 8 − 12 +(− 16 )

)

=

(

− 83

27 − 36

)

When a matrixAis multiplied by another matrixB,
a single matrix results in which elements are obtained
from the sum of the products of the corresponding rows
ofAand the corresponding columns ofB.
Two matricesAandBmay be multiplied together,
provided the number of elements in the rows of matrix
Aare equal to the number of elements in the columns of
matrixB. In general terms, when multiplying a matrix
of dimensions (mbyn) by a matrix of dimensions (nby
r), the resulting matrix has dimensions (mbyr). Thus
a 2 by 3 matrix multiplied by a 3 by 1 matrix gives a
matrix of dimensions 2 by 1.

Problem 5. IfA=

(

23

1 − 4

)

andB=

(

− 57

− 34

)

findA×B.

LetA×B=CwhereC=

(

C 11 C 12

C 21 C 22

)

C 11 is the sum of the products of the first row elements
ofAand the first column elements ofBtaken one at a
time,

i.e. C 11 =( 2 ×(− 5 ))+( 3 ×(− 3 ))=− 19

C 12 is the sum of the products of the first row elements
ofAand the second column elements ofB, taken one
at a time,

i.e. C 12 =( 2 × 7 )+( 3 × 4 )= 26

C 21 is the sum of the products of the second row ele-

ments ofAand the first column elements ofB, taken

one at a time,

i.e. C 21 =( 1 ×(− 5 ))+(− 4 ×(− 3 ))= 7

Finally,C 22 is the sum of the products of the second

row elements ofAand the second column elements of

B, taken one at a time,

i.e. C 22 =( 1 × 7 )+((− 4 )× 4 )=− 9

Thus,A×B=

(

−19 26

7 − 9

)

Problem 6. Simplify

⎛

⎝

340

− 26 − 3

7 − 41

⎞

⎠×

⎛

⎝

2

5

− 1

⎞

⎠

The sum of the products of the elements ofeach row of

the first matrix and the elements of the second matrix,

(called acolumn matrix), are taken one at a time. Thus:

⎛

⎝

340

− 26 − 3

7 − 41

⎞

⎠×

⎛

⎝

2

5

− 1

⎞

⎠

=

⎛

⎝

( 3 × 2 ) +( 4 × 5 ) +( 0 ×(− 1 ))

(− 2 × 2 )+( 6 × 5 ) +(− 3 ×(− 1 ))

( 7 × 2 ) +(− 4 × 5 )+( 1 ×(− 1 ))

⎞

⎠

=

⎛

⎝

26

29

− 7

⎞

⎠

Problem 7. IfA=

⎛

⎝

340

− 26 − 3

7 − 41

⎞

⎠and

B=

⎛

⎝

2 − 5

5 − 6

− 1 − 7

⎞

⎠,findA×B.

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:

⎛

⎝

340

− 26 − 3

7 − 41

⎞

⎠×

⎛

⎝

2 − 5

5 − 6

− 1 − 7

⎞

⎠