Higher Engineering Mathematics, Sixth Edition

The theory of matrices and determinants 237

The inverse of matrix

(

pq

rs

)

is obtained by inter-

changing the positions ofpands, changing the signs
ofqandrand multiplying by the reciprocal of the

determinant

∣

∣

∣

∣

pq

rs

∣

∣

∣

∣. Thus, the inverse of

(

3 − 2

74

)

=

1

( 3 × 4 )−(− 2 × 7 )

(

42

− 73

)

=

1

26

(

42

− 73

)

=

⎛

⎜

⎜

⎝

2

13

1

13

− 7

26

3

26

⎞

⎟

⎟

⎠

Now try the following exercise

Exercise 95 Further problems on the

inverse of 2 by 2 matrices

1. Determine the inverse of

(

3 − 1

− 47

)

⎡

⎢

⎢

⎣

⎛

⎜

⎜

⎝

7

17

1

17

4

17

3

17

⎞

⎟

⎟

⎠

⎤

⎥

⎥

⎦

2. Determine the inverse of

⎛

⎜

⎜

⎝

1

2

2

3

−

1

3

−

3

5

⎞ ⎟ ⎟ ⎠ ⎡ ⎢ ⎢ ⎣

⎛

⎜

⎜

⎝

7

5

7

8

4

7

− 4

2

7

− 6

3

7

⎞

⎟

⎟

⎠

⎤

⎥

⎥

⎦

3. Determine the inverse of

(

− 1. 37. 4

2. 5 − 3. 9

)

⎡

⎣

(

0 .290 0. 551

0 .186 0. 097

)

correct to 3 dec. places

⎤

⎦

22.6 The determinant of a 3 by 3 matrix

(i) Theminorof an element of a 3 by 3 matrix is

the value of the 2 by 2 determinant obtained by

covering up the row and column containing that

element.

Thus for the matrix

⎛

⎝

123

456

789

⎞

⎠the minor of

element 4 is obtained by covering the row

(4 5 6) and the column

⎛

⎝

1

4

7

⎞

⎠, leaving the 2 by

2 determinant

∣

∣

∣

∣

23

89

∣

∣

∣

∣, i.e. the minor of element 4

is( 2 × 9 )−( 3 × 8 )=−6.

(ii) The sign of a minordepends on itspositionwithin

the matrix, the sign pattern being

⎛

⎝

+−+

−+−

+−+

⎞

⎠.

Thus the signed-minor of element 4 in the matrix⎛

⎝

123

456

789

⎞

⎠is−

∣

∣

∣

∣

23

89

∣

∣

∣

∣=−(−^6 )=6.

The signed-minor of an element is called the

cofactorof the element.

(iii) The value of a 3 by 3 determinant is the

sum of the products of the elements and their

cofactors of any row or any column of the

corresponding 3 by 3 matrix.

There are thus six different ways of evaluating a 3× 3

determinant—and all should give the same value.

Problem 14. Find the value of

∣ ∣ ∣ ∣ ∣ ∣

34 − 1

207

1 − 3 − 2

∣ ∣ ∣ ∣ ∣ ∣

The value of this determinant is the sum of the products

of the elements and their cofactors, of any row or of any

column. If the second row or second column is selected,

the element 0 will make the product of the element and

its cofactor zero and reduce the amount of arithmetic to

be done to a minimum.

Supposing a second row expansion is selected.

The minor of 2 is the value of the determinant remain-

ing when the row and column containing the 2 (i.e.

the second row and the first column), is covered up.

Thus the cofactor of element 2 is

∣

∣

∣

∣

4 − 1

− 3 − 2

∣

∣

∣

∣i.e.−11.

The sign of element 2 is minus, (see (ii) above), hence

the cofactor of element 2, (the signed-minor) is+11.

Similarly the minor of element 7 is

∣

∣

∣

∣

34

1 − 3

∣

∣

∣

∣i.e.−13,

and its cofactor is+13. Hence the value of the sum of