272 MATRICES AND DETERMINANTS25.5 The inverse or reciprocal of a
2 by 2 matrix
The inverse of matrixAisA−^1 such thatA×A−^1 =I,
the unit matrix.
Let matrixAbe
(
12
34)
and let the inverse matrix,A−^1 be(
ab
cd)
.Then, sinceA×A−^1 =I,
(
12
34
)
×(
ab
cd)
=(
10
01)Multiplying the matrices on the left hand side, gives
(
a+ 2 cb+ 2 d
3 a+ 4 c 3 b+ 4 d)
=(
10
01)Equating corresponding elements gives:b+ 2 d=0, i.e.b=− 2 dand 3 a+ 4 c=0, i.e.a=−4
3cSubstituting foraandbgives:
⎛⎜
⎝−4
3c+ 2 c − 2 d+ 2 d3(
−4
3c)
+ 4 c 3(− 2 d)+ 4 d⎞⎟
⎠=(
10
01)i.e.(
2
3c 0
0 − 2 d)=(
10
01)showing that2
3c=1, i.e.c=3
2and− 2 d=1, i.e.d=−1
2
Sinceb=− 2 d,b=1 and sincea=−4
3c,a=−2.Thus the inverse of matrix
(
12
34)
is(
ab
cd)
thatis,(
− 21
3
2−1
2)There is, however,a quicker method of obtaining
the inverseofa2by2matrix.
For any matrix(
pq
rs)
the inverse may beobtained by:
(i) interchanging the positions ofpands,
(ii) changing the signs ofqandr, and(iii) multiplying this new matrix by the reciprocal ofthe determinant of(
pq
rs)Thus the inverse of matrix(
12
34)
is1
4 − 6(
4 − 2
− 31)
=(
− 21
3
2−1
2)as obtained previously.Problem 13. Determine the inverse of
(
3 − 2
74)The inverse of matrix(
pq
rs)
is obtained by inter-changing the positions ofpands, changing the signs
ofqandrand multiplying by the reciprocal of thedeterminant∣
∣
∣
∣pq
rs∣
∣
∣
∣. Thus, the inverse of
(
3 − 2
74)
=1
(3×4)−(− 2 ×7)(
42
− 73)=1
26(
42
− 73)
=⎛⎜
⎜
⎝2
131
13
− 7
263
26⎞⎟
⎟
⎠Now try the following exercise.Exercise 110 Further problems on the
inverse of 2 by 2 matrices- Determine the inverse of
(
3 − 1
− 47)⎡⎢
⎢
⎣⎛⎜
⎜
⎝7
171
17
4
173
17⎞⎟
⎟
⎠⎤⎥
⎥
⎦- Determine the inverse of
⎛⎜
⎜
⎝1
22
3−1
3−3
5⎞ ⎟ ⎟ ⎠ ⎡ ⎢ ⎢ ⎣⎛⎜
⎜
⎝75
784
7− 42
7− 63
7⎞⎟
⎟
⎠⎤⎥
⎥
⎦