Matrix Inversion

A.2 Matrix Inversion

The linear system of equations in Eq. (A.3) can be solved by matrix inversion. In the matrix equation AX ⫽ B , we may invert A to get X, i.e.,

X⫽A ⫺ 1 B (A.13)

where ⫺ A 1 is the inverse of A. Matrix inversion is needed in other applications apart from using it to solve a set of equations.

By definition, the inverse of matrix A satisfies

A ⫺ 1 A ⫽ AA ⫺ 1 ⫽I

Appendix A Simultaneous Equations and Matrix Inversion

A-5

where is an identity matrix. ⫺ I A 1 is given by

where adj A is the adjoint of A and det A ⫽ 0A 0 is the determinant of

A . The adjoint of A is the transpose of the cofactors of A. Suppose we are given an n ⫻n matrix A as

The cofactors of A are defined as

where the cofactor c i j is the product of (⫺1) i ⫹j and the determinant of the 1n ⫺ 12 ⫻ 1n ⫺ 12 submatrix is obtained by deleting the

i th row and jth column from A. For example, by deleting the first row and the first column of A in Eq. (A.16), we obtain the cofactor

Once the cofactors are found, the adjoint of A is obtained as

where T denotes transpose. In addition to using the cofactors to find the adjoint of A, they are also used in finding the determinant of A which is given by

0A 0 ⫽ a n

a i j c i j (A.20)

j⫽ 1

where i is any value from 1 to n. By substituting Eqs. (A.19) and (A.20) into Eq. (A.15), we obtain the inverse of A as

0A 0 (A.21)

For a 2⫻2 matrix, if

ab

cd d

A⫽ c (A.22)

A-6

Appendix A Simultaneous Equations and Matrix Inversion

its inverse is

ad ⫺ bc c ⫺c a d 0A 0 (A.23)

1 d c ⫺b ⫺c a d⫽

A ⫺ 1 ⫽ 1 d ⫺b

For a 3⫻3 matrix, if

a 31 a 32 a 33 we first obtain the cofactors as

c 31 c 32 c 33 where

The determinant of the 3⫻3 matrix can be found using Eq. (A.11). Here, we want to use Eq. (A.20), i.e.,

0A0⫽a 11 c 11 ⫹a 12 c 12 ⫹a 13 c 13 (A.27)

The idea can be extended n 73 , but we deal mainly with 2⫻2 and

3⫻3 matrices in this book.

Example A.3

Use matrix inversion to solve the simultaneous equations 2x 1 ⫹ 10x 2 ⫽ 2, ⫺x 1 ⫹ 3x 2 ⫽7

Solution:

We first express the two equations in matrix form as

⫺ AX ⫽ B 1 ¡X⫽A B

2 7 The determinant of A is 0A 0 ⫽ 2 ⫻ 3 ⫺ 10(⫺1) ⫽ 16 , so the inverse

of A is

A ⫺ 1 ⫽ 1 3 ⫺10

16 c 1 2 d

Appendix A Simultaneous Equations and Matrix Inversion

i.e., and x 1 ⫽⫺ 4 x 2 ⫽1 .

Solve the following two equations by matrix inversion.

Practice Problem A.3

2y 1 ⫺y 2 ⫽

4, y 1 ⫹ 3y 2 ⫽9

Answer: y 1 ⫽ 3, y 2 ⫽ 2.

Determine x 1 ,x 2 , and x 3 for the following simultaneous equations using

Example A.4

matrix inversion.

x 1 ⫹x 2 ⫹x 3 ⫽5 ⫺x 1 ⫹ 2x 2 ⫽9 4x 1 ⫹x 2 ⫺x 3 ⫽⫺ 2

Solution:

In matrix form, the equations become

AX ⫽ B ⫺ 1 ¡X⫽A B

We now find the cofactors

c 11 ⫽ 2 2 ⫽ ⫺2, c 12 ⫽⫺ 2 2 ⫽ ⫺1, c 13 ⫽

c 21 ⫽⫺ 2 2 ⫽ 2, c 22 ⫽ 2 c ⫽⫺

c 31 ⫽ 2 2 ⫽ ⫺2, c 32 ⫽⫺

20 2 10 2 ⫽ ⫺1, c

A-8

Appendix A Simultaneous Equations and Matrix Inversion

The adjoint of matrix A is

3 ⫺ 9 3 3 We can find the determinant of A using any row or column of A. Since

one element of the second row is 0, we can take advantage of this to find the determinant as

0A 0 ⫽ ⫺1c 21 ⫹ 2c 22 ⫹ (0)c 23 ⫽⫺ 1(2) ⫹ 2(⫺5) ⫽ ⫺12

Hence, the inverse of A is ⫺ 2 2 ⫺2

i.e., . x 1 ⫽⫺ 1, x 2 ⫽ 4, x 3 ⫽ 2

Practice Problem A.4

Solve the following equations using matrix inversion.

y 1 ⫺y 3 ⫽ 1 2y 1 ⫹ 3y 2 ⫺y 3 ⫽ 1

1 y ⫺y 2 ⫺y 3 ⫽ 3

Answer: y 1 ⫽

6, y 2 ⫽⫺ 2, y 3 ⫽ 5.

Appendix B