The theory of matrices and determinants 237
The inverse of matrix
p q
r s
is obtained by interchanging the positions of p and s, changing the signs
of q and r and multiplying by the reciprocal of the
determinant
p q
r s
. Thus, the inverse of
3 −2
7
4
=
1
(3 × 4) − (−2 × 7)
4 2
−7 3
=
1
26
4 2
−7 3
=
⎛
⎜
⎜
⎝
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
−4
7
⎡
⎢
⎢
⎣
⎛
⎜
⎜
⎝
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
−1.3
7.4
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) The minor of 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
⎛
⎝
1 2 3
4 5 6
7 8 9
⎞
⎠ 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
2 3
8 9
, i.e. the minor of element 4
is (2 × 9) −(3 × 8) =−6.
(ii) The sign of a minor depends on its position within
the matrix, the sign pattern being
⎛
⎝
+ − +
− + −
+ − +
⎞
⎠ .
Thus the signed-minor of element 4 in the matrix
⎛
⎝
1 2 3
4 5 6
7 8 9
⎞
⎠ is −
2 3
8 9
=−(−6) = 6.
The signed-minor of an element is called the
cofactor of 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
3
4 −1
2
0
7
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 remaining 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
3
4
1 −3
i.e. −13,
and its cofactor is +13. Hence the value of the sum of
The inverse of matrix
p q
r s
is obtained by interchanging the positions of p and s, changing the signs
of q and r and multiplying by the reciprocal of the
determinant
p q
r s
. Thus, the inverse of
3 −2
7
4
=
1
(3 × 4) − (−2 × 7)
4 2
−7 3
=
1
26
4 2
−7 3
=
⎛
⎜
⎜
⎝
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
−4
7
⎡
⎢
⎢
⎣
⎛
⎜
⎜
⎝
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
−1.3
7.4
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) The minor of 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
⎛
⎝
1 2 3
4 5 6
7 8 9
⎞
⎠ 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
2 3
8 9
, i.e. the minor of element 4
is (2 × 9) −(3 × 8) =−6.
(ii) The sign of a minor depends on its position within
the matrix, the sign pattern being
⎛
⎝
+ − +
− + −
+ − +
⎞
⎠ .
Thus the signed-minor of element 4 in the matrix
⎛
⎝
1 2 3
4 5 6
7 8 9
⎞
⎠ is −
2 3
8 9
=−(−6) = 6.
The signed-minor of an element is called the
cofactor of 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
3
4 −1
2
0
7
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 remaining 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
3
4
1 −3
i.e. −13,
and its cofactor is +13. Hence the value of the sum of