082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

8.6.1 Findingtheinverseofamatrix
For 2 ×2 matrices a simple formula exists tofind the inverse of
( ) a b
A =
c d
This formulastates
IfA=
( ) a b
thenA
c d
−1 =
1
ad−bc
8.6 The inverse of a 2 × 2 matrix 275
( ) d −b
.
−c a
Example8.20 IfA =
( ) 3 5
findA
1 2
−1 .
Solution Clearlyad −bc = 6 −5 = 1,so that
A −1 = 1 ( ) ( ) 2 −5 2 −5
=
1 −1 3 −1 3
The solution should always be checked by formingAA −1 .
Example8.21 IfA =
( ) 1 5
findA
2 4
−1 .
Solution Herewehavead −bc = 4 −10 = −6.Therefore
⎛
A −1 = 1 ( ) − 2 5
⎞
4 −5 ⎜
=
3 6⎟
−6 −2 1
⎝
1
⎠
3 −1 6
Example8.22 IfA =
( ) 2 4
findA
1 2
−1 .
1
Solution Thistime,ad −bc = 4 −4 = 0,sowhenwecometoform
ad−bc wefind1/0which
isnotdefined. We cannot form the inverse ofAinthiscase; itdoes notexist.
Clearly not all square matrices have inverses. The quantity ad − bc is obviously the
important determining factor since only if ad −bc ≠ 0 can we find A −1 . This quantity
is therefore given a special name: the determinant ofA, denoted by |A|, or detA.
Given any 2 × 2 matrix A, its determinant, |A|, is the scalar ad − bc. This is easily
remembered as
[product of ց diagonal] −[product of ւ diagonal]