Some examples for identity matrixes are I1 = , I2 = ,

I3 = …

Let’s analyze the multiplication of any matrix with identity matrix

. =

. =

We can conclude that I1, I2, I3, I4… are multiplicative identity matrix.

Pay attention that (I m)n = I m

Definition: A-1 is called the multiplicative inverse matrix for any

square matrix A if A.A-1= A-1.A = I

Inverse of Matrix (2×2):

Let’s find the multiplicative inverse of the matrix

A= 2 51 3

The other way to find the inverse matrix of

is

assume that

a bA

c d

a bA

c d

1 x yA

z t

We also notice that A-1 exists provided ad – bc ≠ 0, otherwise

would be undefined . If ad – bc ≠ 0, we say that A is invertible.

Now, find the multiplicative inverse of A= , 10 25 1

B

Example:1. If A= is not invertible, what is the value of x?

Example:

is not invertible. Find the value of x. sin cos

cos sin2 2

x xA x x

Example: and are givens. Find the value of x.

2 0xA

x x

1 1 01 2

A

Example: A= is given. If the inverse of the

matrix A is equal to the matrix A find the value of a.

2 21 4

A

Example:

and are givens. Find

which justifies A.X=B

51

B

aX

b

If A.X=B, how can we find X using matrices only?

By using the definition,………………………………………………………

………………………………………………………

………………………………………………………

With the same idea X.A = B => (X.A).A-1=B.A-1

X(A.A-1)= B.A-1

X=B.A-1

Other properties of inverse:• (A.B)-1 = B-1.A-1

• (A-1) -1 =A

• (An)-1 = (A-1)n

Example: A= and B=

Find (A-1.B)-1

Example:

If B . C-1 = A then find the matrix C.

0 1 1 0 , B=

1 2 1 2A

Example:

If

1 01 2

A