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8 Inverse Matrix

In this section of we will examine two methods of finding the inverse of a matrix, these are

•The adjoint method.

•Gaussian Elimination.

8.1 Matrix Inverse: The Adjoint Method

We require a couple of definitions before we set out the procedure to find the inverse of a matrix.

8.1.1 Type of Matrix: Cofactor matrix

Definition 8.1(Cofactor Matrix).

Given an×nmatrixA. The cofactor matrixCofAis the matrix formed by evaluating the cofactors of each entry inA C=( (C

11C12... C1n

C

21C22... C2n.

C n1Cn2... Cnn) Example 8.1.1(Cofactor Matrix).Find the cofactor matrix for A=( (-1 1 0 2 0 0

1 1-2)

Solution:

In order to find the cofactor matrix forAwe will need the cofactors of each and every entry inA, C

11= (-1)1+1M11= (-1)2det?0 0

1-2? = 0 C

12= (-1)1+2M12= (-1)3det?2 0

1-2? = 4 C

13= (-1)1+3M13= (-1)4det?2 01 1?

= 2 41
continuing this process (you should check this) we will find C

21= 2, C22= 2, C23= 2

C

31= 0, C32= 0, C33=-2

and thus the cofactor matrix is C=( (0 4 22 2 20 0-2)

8.1.2 Adjoint of a matrix

Definition 8.2(Adjoint of a matrix).

The adjoint of a matrixAdenoted adj(A) is simply thetranspose of the of the cofactor matrix. That is, ifCdenotes the cofactor matrix ofAthen adj(A) =C?

Example 8.1.2(The adjoint).

Find the adjoint of the matrix

A=( (-1 1 0 2 0 0

1 1-2)

Solution:

We have done all the hard work (finding the cofactor matrix) inthe previous example C=( (0 4 22 2 20 0-2) thus, adj(A) =C?=( (0 2 04 2 02 2-2) 42

8.1.3 The Inverse: Using the adjointWe are now ready to state (without proof) a useful theorem which will allow us to com-

pute the inverse of a matrix.

Theorem 8.1.1(Inverse using the adjoint).

LetAbe an×nmatrix. If detA?= 0, then

A -1=1 detAadjA The steps involved in finding an inverse using an the adjoint method for a matrixA

1. Find the determinant of the matrix of interest detA

•If detA?= 0 then the inverse will exist.

•If detA= 0 or matrix isn"t square then the inverse will not exist.

2. Find the cofactor matrixC, by finding the cofactor for each element ofA.

•The cofactor of theith-rowjth-column element ofAis C ij= (-1)i+jMij whereMijis the minor.

3. Find the adjoint ofA

adjA=C?

4. The inverse is given by

A -1=1 detAadjA

Example 8.1.3(The Inverse).

Find the inverse of

A=( (-1 1 0 2 0 0

1 1-2)

using the adjoint method.

Solution:

43

We have the cofactor matrix and the adjoint ofA

C=( (0 4 22 2 20 0-2) )and adj(A) =( (0 2 04 2 02 2-2) We can find the determinant ofAby performing a cofactor expansion about any row or column ofA. Picking the third column (as it has two zeros) we have detA=a13C13+a23C23+a33C33 we have all the cofactors (from the cofactor matrix) thus, detA= (0)(2) + (0)(2) + (-2)(-2) = 4. According to our theorem concerning the adjoint and the inverse of a matrix we have A -1=1 detAadjA=14( (0 2 04 2 02 2-2) and thus, A -1=( (0 1/20 1 1/20

1/21/2-1/2)

We can check if this is in fact the inverse

AA -1=( (-1 1 0 2 0 0

1 1-2)

(0 1/20 1 1/20

1/21/2-1/2)

(1 0 00 1 00 0 1) and A -1A=( (0 1/20 1 1/20

1/21/2-1/2)

(-1 1 0 2 0 0

1 1-2)

(1 0 00 1 00 0 1) and thus we have an inverse. 44

8.2 Matrix Inverse: Gaussian Elimination MethodAnother useful method used to find an inverse of matrix involvessubjecting our matrix

to a series ofelementary row operations.

8.2.1 Operation: Elementary Row Operations

There are three types of elementary tow operations

1. Add/subtract a multiple of one row to another row.

2. Multiply a row by a constant.

3. Interchange two rows.

Interestingly these elementary row operations have very specific effects on the determi- nant of a matrix.

Row OperationEffect on determinant

Add a multiple of one row to another rowNone

Multiply a row by a constantkmultiplied byk

Interchange two rowsmultiplied by-1.

How can this be used to find a determinant for matrix? We can reduce a matrixA to upper triangular form using elementary row operations making it a matrixA?. The determinant ofA?is easy to find (as it is triangular the determinant is simply the product of the entries on the diagonal) and relate its determinant to the determinant ofAby working back through the row operations that were used in thereduction process. Example 8.2.1(The determinant using elementary row operations).

Find the determinant of

A=( (2 4 91 2 41 10 7) using elementary row operations.

Solution:

(2 4 91 2 41 10 7) )R3 to R3-R2---------→(det unchanged)( (2 4 91 2 40 8 3) )swap R2 and R3---------→(det× -1)( (2 4 90 8 31 2 4) )2×R3-----→(det×2)( (2 4 90 8 32 4 8)

R3 to R3-R1---------→(det unchanged)(

(2 4 90 8 30 0-1) 45
We now have the matrixAtransformed into an upper triangular matrix A (2 4 90 8 30 0-1) the determinant ofA" is given by the product of the elements on the diagonal detA?= (2)(8)(-1) =-16 The operations that we conducted on the matrixAwere

Row OperationEffect on determinant

Add a multiple of one row to another rowdet unchanged

Interchange two rowsmultiplied det by-1

Multiply a row by 2multiplied det by 2

Add a multiple of one row to another rowdet unchanged and thus, detA?= (-1)(2)detA thus, detA= 8.

8.2.2 Matrix inverse using row operations

We can use these row operations to find the inverse of a matrix, the result that we will use is quoted here without proof. If a sequence of elementary row operations on a square matrixAcan reduce the matrix to the identity matrixI, then the same sequence of row operations applied toIwill result inIbeing transformed toA-1.

Of note is that

•If it"s not possible to reduceAtoIusing elementary row operations thenAis not invertible. •IfAis invertible then there will be more than one way to reduce ittoI. Since we are going to perform the same operations on a given matrix A=( (a

11a12a13

a

21a22a23

a

31a32a33)

)andI=( (1 0 00 1 00 0 1) 46
We will introduce the followingaugmented matrix, which will allow us to manipulate both matrices at the same time easily (a

11a12a13

1 0 0 a

21a22a23

0 1 0 a

31a32a33

0 0 1)

which is nothing more than both the matrices placed adjacentto one another. Example 8.2.2(Inverse using row operations and an augmented matrix).

Find the inverse of

A=( (0 1 21 2 02 0 1) using elementary row operations.

Solution:

Step 1:Augment the matrix with the identity matrix (0 1 2 1 0 0 1 2 0 0 1 0 2 0 1

0 0 1)

Step 2:Swap rows (and multiply by a constant if necessary) to ensurethat the left side of the augmented matrix will have a "1" in the first row first column entry (0 1 2quotesdbs_dbs20.pdfusesText_26