Matrix Division

CW

We have seen that for 2x2 (“two by two”) matrices A and B then AB BA

To divide matrices we need to define what we mean by division!

Matrix Division

CW

We have seen that for 2x2 (“two by two”) matrices A and B then AB BA

To divide matrices we need to define what we mean by division!

With numbers or algebra we use b/a to solve ax=b. The equivalent in 2x2 matrices is to solve AX=B where A, B and X are 2x2 matrices.

Identity MatrixCW

With numbers or algebra we use b/a to solve ax=b. The equivalent in 2x2 matrices is to solve AX=B where A, B and X are 2x2 matrices.

We first need to define the identity matrix - the matrix I for which IX = XI = X for all X (For multiplying number the identity is

Identity MatrixCW

With numbers or algebra we use b/a to solve ax=b. The equivalent in 2x2 matrices is to solve AX=B where A, B and X are 2x2 matrices.

We first need to define the identity matrix - the matrix I for which IX = XI = X for all X (For multiplying number the identity is 1).

The identity 2x2 matrix is

Identity MatrixCW

With numbers or algebra we use b/a to solve ax=b. The equivalent in 2x2 matrices is to solve AX=B where A, B and X are 2x2 matrices.

We first need to define the identity matrix - the matrix I for which IX = XI = X for all X (For multiplying number the identity is 1).

The identity 2x2 matrix is

10

01

Identity MatrixCW

With numbers or algebra we use b/a to solve ax=b. The equivalent in 2x2 matrices is to solve AX=B where A, B and X are 2x2 matrices.

We first need to define the identity matrix - the matrix I for which IX = XI = X for all X (For multiplying number the identity is 1).

The identity 2x2 matrix is

The identity 3x3 matrix is

10

01

100

010

001

Identity MatrixCW

We first need to define the identity matrix - the matrix I for which IX = XI = X for all X (For multiplying number the identity is 1).

The identity 2x2 matrix is

The identity 3x3 matrix is

In general if X is an mxn matrix then ImX = XIn = X

10

01

100

010

001

Identity MatrixCW

We first need to define the identity matrix - the matrix I for which IX = XI = X for all X (For multiplying number the identity is 1).

The 2x2 identity matrix (I2) is

The 3x3 identity matrix (I3)is

In general if X is an mxn matrix then ImX = XIn = X

10

01

100

010

001

Inverse MatrixCW

In numbers, the inverse of 3 is 1/3 = 3-1

In algebra, the inverse of a is 1/a = a-1

In matrices, the inverse of A is A-1

Inverse MatrixCW

In numbers, the inverse of 3 is 1/3 = 3-1

In algebra, the inverse of a is 1/a = a-1

In matrices, the inverse of A is A-1

3-1 is defined so that 3x 3-1 = 1a-1 is defined so that a x a-1 = 1A-1 is defined so that A A-1 = I

Inverse MatrixCW

In numbers, the inverse of 3 is 1/3 = 3-1

In algebra, the inverse of a is 1/a = a-1

In matrices, the inverse of A is A-1

3-1 is defined so that 3 x 3-1 = 3-1 x 3 = 1a-1 is defined so that a x a-1 = a-1 x a = 1A-1 is defined so that A A-1 = A-1 A = I

However, for a square matrix A there is not always an inverse A-1

Inverse MatrixCW

In matrices, the inverse of A is A-1

A-1 is defined so that A A-1 = A-1 A = I

However, for a square matrix A there is not always an inverse A-1

If A-1 does not exist then the matrix is said to be singular

If A-1 does exist then the matrix is said to be non-singular

Inverse MatrixCW

In matrices, the inverse of A is A-1

A-1 is defined so that A A-1 = A-1 A = I

If A-1 does not exist then the matrix is said to be singular

If A-1 does exist then the matrix is said to be non-singular

A square matrix A has an inverse if, and only if, A is non-singular.

Inverse MatrixCW

In matrices, the inverse of A is A-1

A-1 is defined so that A A-1 = A-1 A = I

A square matrix A has an inverse if, and only if, A is non-singular.

If A-1 does exist the the solution to AX=B is

X = A-1 B

Inverse MatrixCW

A-1 is defined so that A A-1 = A-1 A = I

If A-1 does exist the the solution to AX=B is

AX = BPre-multiply by A-1 A-1AX = A-1B

Inverse MatrixCW

A-1 is defined so that A A-1 = A-1 A = I

If A-1 does exist the the solution to AX=B is

AX = BPre-multiply by A-1 A-1AX = A-1B

But A-1A = I so IX = A-1B X = A-1B

Inverse MatrixCW

AX = BPre-multiply by A-1 A-1AX = A-1B

But A-1A = I so IX = A-1B X = A-1B

If the inverse of A is A-1 then the inverse of A-1 is A. This is because if AC = I then CA = I, and also any matrix inverse is unique.

Inverse MatrixCW

If the inverse of A is A-1 then the inverse of A-1 is A. This is because if AC = I then CA = I, and also any matrix inverse is unique.

What is the inverse of

30

02A

Inverse MatrixCW

If the inverse of A is A-1 then the inverse of A-1 is A. This is because if AC = I then CA = I, and also any matrix inverse is unique.

What is the inverse of

30

02A

y

x

0

0let 1A

Inverse MatrixCW

If the inverse of A is A-1 then the inverse of A-1 is A. This is because if AC = I then CA = I, and also any matrix inverse is unique.

What is the inverse of

Then solve for u, v, w, x

30

12B

xw

vu1let B

20

13

6

11B

General Inverse MatrixCW

If the inverse of A is A-1 then the inverse of A-1 is A. This is because if AC = I then CA = I, and also any matrix inverse is unique.

What is the inverse of

dc

baC

General Inverse MatrixCW

If the inverse of A is A-1 then the inverse of A-1 is A. This is because if AC = I then CA = I, and also any matrix inverse is unique.

What is the inverse of

dc

baC

xw

vu1let C

Then solve for u, v, w, x

General Inverse MatrixCW

dc

baC

bcadD

ac

bd

Dxw

vu

where

1let 1C

1

0

0

1

dxcv

bxav

dwcu

bwau

a

c

cwbcad

Subtract

dawcau

cbcwacu

)(

:

0

General Inverse MatrixCW

What is the inverse of

363

121

3

1

63

21

3

1

63

21

yx

yx

yx

yx

y

x

xw

vu1let C

Then solve for u, v, w, x