The Chemistry Maths Book, Second Edition

520 Chapter 18Matrices and linear transformations

so that Afollowed by Bis equivalent to the single transformation whose matrix

representation is the matrix productC 1 = 1 BA.

EXAMPLE 18.20Consecutive transformations in two dimensions

Let(x, y)be the cartesian coordinates of a point in the xy-plane, and let the matrices

represent transformations in the plane. Matrix Arepresents anticlockwise rotation

throughπ 23 about the origin, Bis reflection in the linex 1 = 1 y, and Cis inversion

through the origin. The sequence Afollowed by Bfollowed by Cis illustrated in

Figure 18.5, and is equivalent to the single transformation

The final position of the point is

0 Exercise 60

Inverse transformations

If A is a nonsingular square matrix then it has the unique inverseA

− 1

such that

A

− 1

Ax 1 = 1 AA

− 1

x 1 = 1 Ix 1 = 1 x (18.53)

()′′′, ′′′ =− − ,− +.

















xy x y x y

3

2

1

2

1

2

3

2

DCBA==

−

−

































−





10

01

01

10

1

2

3

2

3

2

1

2

























=

−−

−





























3

2

1

2

1

2

3

2

ABC=

−





























,=

















,=

1

2

3

2

3

2

1

2

01

10

−−

−

















10

01

...

..

...

...

..

...

...

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θ

• 
  


• 
  

(x,y)

(x

′

,y

′

)

o

y

.......

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• 
  


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′′

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′′

)

(x

′

,y

′

)

x=y

o

y

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B

• 
  


• 
  

(x

′′

,y

′′

)

(x

′′′

,y

′′′

)

o

x

y

.......

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C

Figure 18.5