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) --------------   