MAT 2401Linear Algebra

2.3 The Inverse of a Matrix

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HW

WebAssign 2.3 Written HW

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The definition of the Inverse of a matrix.

Formula of the inverse for 2x2 matrices.

Use row operations to find the inverse of nxn matrices.

Recall

4

2 2 5 11

4 6 8 24

1 1 1 4

2 2 5 , , 11

4 6 8 24

?

x y z

x y z

x y z

x

A X y b

AX b

X

z

can be represented by

where

Can we "divide" both side with A to fi nd the solution

Recall Identity Matrix

nxn Square Matrix

1 0 0

0 1

0

0 0 1

nI I

Inverse Matrix

Let A be a square matrix. Then the inverse for A is a square matrix A-1 of the same size as A such that

AA-1 = I = A-1A

Inverse Matrix

Let A be a square matrix. Then the inverse for A is a square matrix A-1 of the same size as A such that

AA-1 = I = A-1A If such inverse A-1 exists, then the

matrix A is said to be invertible (otherwise, singular).

Example 1 (a)2 1 1 1

and 1 1 1 2

2 1 1 1

1 1 1 2

1 1 2 1

1 2 1 1

A B

AB

BA

Inverse of a 2x2 Matrix

1

If ,

1then

provided that ____________

a bA

c d

d bA

c aad bc

Example 1 (b)

12 1 Find

1 1A A

1 1 d bA

c aad bc

Matrix Equations

AX b

Example 1 (c)2 5

3

x y

x y

Use matrix inverse to solve

1 1 1 1

1 2X A b A

Example 1 (c)2 5

3

x y

x y

Use matrix inverse to solve

1 1 1 1

1 2X A b A

Remarks

When n≥3, there are no useful formula to find the inverse of an invertible matrix.

How to Find A-1 ?

12 1Let ,

1 1

2 1 1 0So,

1 1 0 1

Then,

2 1

1 1

2 1and

1 1

A A

1 1 1

1 2A

Example 1 (d)

Use row operations to find the inverse of

2 1

1 1A

Inverse of an 3x3 Matrix(Same for nxn matrices) Given matrix A,

we set up the following matrix11 12 13

21 22 23

31 32 33

a a a

A a a a

a a a

Ro11 12 13

21 22 23

31 32 3

w Operation

3

s

1 0 0 1 0 0 * * *

0 1 0 0 1 0 * * *

0 0 1 0 0 1 * * *

a a a

a a a

a a a

Inverse of an 3x3 Matrix(Same for nxn matrices) Given matrix A,

Use row operations to get to the second matrix. A-1 (if exists) is the matrix on the right half.11 12 13

21 22 23

31 32 33

a a a

A a a a

a a a

Ro11 12 13

21 22 23

31 32

w Operations

33

* * *

* * *

*

1 0 0 1 0 0

0 1 0 0 1 0

*0 0 1 0 0 1 *

a a a

a a a

a a a

Example 2 (a)

Find the inverse of 1 1 1

2 2 5

4 6 8

A

Example 2 (b)4

2 2 5 11

4 6 8 24

1 1 1 4

2 2 5 , , 11

4 6 8 24

x y z

x y z

x

AX b

AX b

y z

x

A X y b

z

can be represented by

where

Solve

Example 2 (b)

1

7 1 1

3 3 2 42 2 1

113 3 2

242 1

03 3

X A b

Q&A

We can use two methods to solve a system of equations. (a) Gauss-Jordan Elimination (b) Matrix InverseQ: Why use (b) when (a) is easier?A:

Properties of Matrix Inverses

If A be an invertible matrix, kZ+ , c≠0 is a scalar, then A-1, Ak, cA, and AT are invertible. Also,1. (A-1)-1= A2. (Ak)-1 =(A-1)k

3. (cA)-1 = A-1

4. (AT)-1 =(A-1)T

5. (AB)-1 =B-1A-1

Cancellation Properties

If C is invertible, then1. AC=BC implies A=B2. CA=CB implies A=B