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Definition of an Inverse of a Matrix

Assuming that we have a square matrix A, which is non-singular (i.e. det (A) does not equal zero), then there exists an n × n matrix A-1 which is called the inverse of A such that:
AA-1 = A-1A = I, where I is the identity matrix.

The inverse of a 2×2 matrix
Take for example an arbitrary 2×2 Matrix A whose determinant (ad − bc) is not equal to zero.

where
a, b, c and d are numbers.

The inverse is:


The inverse of a general n × n matrix A can be found by using the following equation.

where the adj (A) denotes the adjoint of a matrix. It can be calculated by the following method:
Given the
n × n matrix A, define B = bij to be the matrix whose coefficients are found by taking the determinant of the (n-1) × (n-1) matrix obtained by deleting the ith row and jth column of A.

The terms of
B (i.e. B = bij) are known as the cofactors of A.
Define the matrix
C, where cij = (−1)i+j bij.
 
The transpose of
C (i.e. CT) is called the adjoint of matrix A.


Example 1: . Find the adj A.
Solution:
Computation of adj A:
Cofactor of 1 = a
11 = - 4
Cofactor of 3 = a
12 = -1
Cofactor of 7 = a
13 = 6
Cofactor of 4 = a
21 = 11
Cofactor of 2 = a
22 = -6
Cofactor of 3 = a
23 = 1
Cofactor of 1 = a
31 = -5
Cofactor of 2 = a
32 =-25
Cofactor of 1 = a
33 = -10
Therefore we have:



Example 2: Find the inverse of

Solution:
The following method to find the inverse is only applicable for 2 × 2 matrices.

1. Interchange leading diagonal elements:
-7 → 2; 2 → -7



2. Change signs of the other 2 elements:
-3 → 3; 4 → -4



3. Find the determinant |A|


4. Multiply result of [2] by 1/ |A

 
 
Example 3: Find the inverse of

Solution:
The cofactor matrix for A can be calculated as follows:
Cofactor of 1 = a11 = 24
Cofactor of 2 = a
12 = 5
Cofactor of 3 = a
13 = -4
Cofactor of 0 = a
21 = -12
Cofactor of 4 = a
22 = 3
Cofactor of 5 = a
23 = 2
Cofactor of 1 = a
31 = -2
Cofactor of 0 = a
32 = -5
Cofactor of 6 = a
33 = 4
So the cofactor of

Therefore, the adjoint of
.

And finally, the inverse of
A is given by,


Example 4: Compute the inverse of

Solution: The cofactor matrix for A can be calculated as follows:
Cofactor of 3: a
11 = 12
Cofactor of 2: a
12 = 6
Cofactor of -1: a
13 = -16
Cofactor of 1: a
21 = 4
Cofactor of 6: a
22 = 2
Cofactor of 3: a
23 = 16
Cofactor of 2: a
31 = 12
Cofactor of -4: a
32 = -10
Cofactor of 0: a
33 = 16

So the cofactor of

Therefore the adjoint of .

And finally, the inverse of
A is given by:


Example 5: Find the inverse of
Solution: Write
Since

We have:

a + c = 1
-a + 2c = 0
b + d = 0
-b + 2d = 1
or
a = 2/3
b= -1/3
c=1/3
d= 1/3

The inverse of A is therefore:

We know that the inverse matrix is unique when it exists. So if A is invertible, then A-1 is also invertible and (A-1)-1 = A.



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