Some examples for identity matrixes are I 1 =, I 2 =, I 3 = … Let’s analyze the multiplication of any matrix with identity matrix. = We can conclude that I 1, I 2, I 3, I 4 … 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=

The other way to find the inverse matrix of is assume that

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=,

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

Example: is not invertible. Find the value of x.

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

Example: A= is given. If the inverse of the matrix A is equal to the matrix A find the value of a.

Example: and are givens. Find which justifies A.X=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 (A n ) -1 = (A -1 ) n

Example: A= and B= Find (A -1.B) -1

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

Example: If