the inverse of a matrix prepared by vince zaccone for campus learning assistance services at ucsb

The Inverse of a Matrix

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

The inverse of a square matrix A is another matrix with the following properties:

IAAAA 11

Here I represents the identity matrix of the same size as A and A-1.

Note that A-1 must be a square matrix of the same size as A.




IAAAA 11

Here is a system of linear equations. To solve it, we can put it into matrix format and try to find the inverse of the coefficient matrix.

Let’s see how that works.4zy2x

5yx

2zyx






IAAAA 11

Here is a system of linear equations. To solve it, we can put it into matrix format and try to find the inverse of the coefficient matrix.

Let’s see how that works.4zy2x

5yx

2zyx





4

5

2

b;

z

y

x

x;

121

011

111

A

bxA

To find the inverse, for an augmented matrix with the coefficient matrix on the left, and the corresponding identity matrix on the right.

Next, row reduce until you have the identity on the left, and the inverse will be on the right.

1AIreduction

rowIA





1AIreduction

rowIA

Here is the method, applied to our example:



100

010

001

121

011

111



1AIreduction

rowIA




13*3

12*2

RRR

RRR

100

010

001

121

011

111

101

011

001

230

120

111



1AIreduction

rowIA




13*3

12*2

RRR

RRR

100

010

001

121

011

111

32*3 R2R3R101

011

001

230

120

111



1AIreduction

rowIA




13*3

12*2

RRR

RRR

100

010

001

121

011

111

32*3 R2R3R101

011

001

230

120

111

231

011

001

100

120

111



1AIreduction

rowIA




13*3

12*2

RRR

RRR

100

010

001

121

011

111

32*3 R2R3R101

011

001

230

120

111

32*2 RRR

231

011

001

100

120

111



1AIreduction

rowIA




13*3

12*2

RRR

RRR

100

010

001

121

011

111

32*3 R2R3R101

011

001

230

120

111

32*2 RRR

231

011

001

100

120

111

231

242

001

100

020

111



1AIreduction

rowIA




13*3

12*2

RRR

RRR

100

010

001

121

011

111

32*3 R2R3R101

011

001

230

120

111

32*2 RRR

231

011

001

100

120

111

22

1*2 RR

231

242

001

100

020

111

231

121

001

100

010

111



1AIreduction

rowIA




13*3

12*2

RRR

RRR

100

010

001

121

011

111

32*3 R2R3R101

011

001

230

120

111

32*2 RRR

231

011

001

100

120

111

22

1*2 RR

231

242

001

100

020

111

231

121

001

100

010

111 321*1 RRRR

1AI

231

121

111

100

010

001

So now we have the inverse of our coefficient matrix. To solve the original system of equations, simply multiply through by this inverse matrix:



bAx

bAxAA

bxA

1

11




231

121

111

121

011

111

z

y

x

4

5

2

231

121

111

bAx

bAxAA

bxA

1

11

z

y

x




231

121

111

121

011

111

z

y

x

4

5

2

231

121

111

bAx

bAxAA

bxA

1

11

5

4

1

z

y

x

Thus we find a unique solution to the original system of equations.

2x2 Inverse Matrix Shortcut

• Theorem 4: Let . If , then

A is invertible and

If , then A is not invertible.• The quantity is called the determinant of A,

and we write • This theorem says that a matrix A is invertible

if and only if det .

a bA

c d

0ad bc

1 1 d bA

c aad bc

0ad bc ad bcdet A ad bc

2 20A


IAAAA 11

Here is a system of linear equations. Notice that the coefficient matrix is the same as the one we solved earlier. We can use the same inverse matrix to solve this one.





4zy2x

3yx

1zyx

4

3

1

b;

z

y

x

x;

121

011

111

A

bxA


IAAAA 11

Here is a system of linear equations. Notice that the coefficient matrix is the same as the one we solved earlier. We can use the same inverse matrix to solve this one.





4zy2x

3yx

1zyx

4

3

1

b;

z

y

x

x;

121

011

111

A

bxA

1A

231

121

111

b

4

3

1

0

1

2

z

y

x

f. The linear transformation is one-to-one.

g. The equation has at least one solution for each b in Rn .

h. The columns of A span Rn .

i. The linear transformation maps Rn onto Rn .

j. There is an matrix C such that .

k. There is an matrix D such that .

l. AT is an invertible matrix.

• Theorem 8: Let A be a square matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true or all false.

a. A is an invertible matrix.

b. A is row equivalent to the identity matrix.

c. A has n pivot positions.

d. The equation has only the trivial solution.

e. The columns of A form a linearly independent set.



n n

x 0A

x xAx bA

x xAn nn n

CA IAD I

the inverse of a matrix prepared by vince zaccone for campus learning assistance services at ucsb

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