# CHE 311 Lecture Notes - Lecture 29: Surface Tension, Viscosity, List Of Forgotten Realms Nations

by OC2140094

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Lecture 4

Inverse of

A

is defined as a matrix

B

such that

BA I

=

or

AB I

=

Prove. If inverse exists, it is unique.

If it is not unique, assume that there is a matrix

C

such that

CA I=

( )

0B CA−=

0

DA=

0

,

DA

rrrn≥+−

,

A

rn=

0

D

r≥

00

D

rD→=→=

,BC=

Q.E.D

We know that,

1

1 A adj A

A

−=

Suppose the rank of

A

is

1n−

A

is singular

0A→=

0Aadj A =

Apply Sylvester’s Law of Nullity,

( )

01

adj A

n rn≥ −+ −

1

adj A

r≤

01

adj A

r→≤ ≤

Only pages 1-2 are available for preview. Some parts have been intentionally blurred.

© 2017 Meenesh R. Singh (mrsingh@uic.edu)

if

0

adj A

r= →

Every cofactor is zero but

10

A

rn= −≠

.

At least one cofactor is non-zero

1

adj A

r→=

adj A

has at least one non-zero column vector if there are more than one column of non-zero

vector, they must be linearly dependent.

Any non-zero column of

adj A

is a solution to the homogenous eq:

0Ax=

.

Let

A

has a rank

A

r

. We want to solve

0Ax=

.

11 1 12 2 1 1, 1 1 1 1

0

rr r r n n

ax ax ax a x ax

++

+ +…+ + +…+ =

21 1 22 2 2 2, 1 1 2 0

rr r r nn

ax ax ax a x ax

++

+ +…+ + +…+ =

11 2 2 , 1 1 0

r r rr r r r r rn n

ax ax ax a x ax

++

+ +…+ + +…+ =

11 2 2 , 1 1

0

n n nr r n r r nn n

ax ax ax a x ax

++

+ +…+ + +…+ =

we are solving for upper left hand corner.

12

,,,

r

xx x…

are unknowns.

12

, ,,

rr n

xx x

++

…

can be assumed.

1st solution

Assume

1

1

r

x

+

=

0, 2, ,

j

x jr n= =+…

Solve,

()() () ( )

11 1

112

, , , ,1 , 0, , 0 T

r

x xx x

= ……

2nd solution

( ) ( ) ( ) ( )

22 2

212

, , , , 0,1 , 0, , 0

T

r

x xx x

= ……

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