Inverse Matrix


Inverse Matrix 


Identity Matrix
Originally we have the identity property for a real number m, where

m  1 = m and 1  m = m

for any real number m. For a square matrix M, the identity property is found by using a square identity matrix I, such that

M  I = M and I  M = M

where I and M must be square matrices of the same size.

Example
Verify the identity property for matrices using matrix M


M  I = M 

Therefore


Then we have each of the elements of the product:

(3  1) + (4  0) + (1  0) = 3 
(3  0) + (4  1) + (1  0) = 4 
(3  0) + (4  0) + (1  0) = 1 

( 1) + (5  0) + (1  0) = 
( 0) + (5  1) + (1  0) = 5 
( 0) + (5  0) + (1  1) = 1 

(7  1) + (1  0) + (0  0) = 7 
(7  0) + (1  1) + (0  0) = 1 
(7  0) + (1  0) + (0  1) = 0 

The resultant matrix is
 
The n x n identity matrix is


Multiplicative Inverses
If M is an n 
 n, matrix, its multiplicative inverse n  n, has to satisfy the following properties: 

MM -1 = In and M -1M = In 

understanding two conditions, where

1.) Only a square matrix can have multiplicative inverse
2.) Even though m-1 = 1/m for any nonzero real number, M -1 
 1/M

Example
Find the inverse of matrix M:

 

we can say that the inverse of matrix M -1 would look like this:


Then using the inverse property we have

MM -1 = In 

 

3a + 4b + c = 1 
3d + 4e + f = 0 
3g + 4h + k = 0 

-2a + 5b + c = 0 
-2d + 5e + f = 1
-2g + 5h + k = 0 

7a + b = 0 
7d + e = 0 
7g + h = 1

By solving for each of the variables we find the inverse matrix M -1:

  

Exercises

If possible find the inverse matrix for the following:
1) 
 


2) 



3) 



4) 






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Answer Key

1) 


2) Not possible since M is not a square matrix

3) 


4)