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
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