❖ In this video, we have explained how to find the inverse of a Matrix and solve a Linear System

❖ To find the inverse of a Matrix, we have mentioned two ways to answer. Then, we used the inverse of a Matrix to solve a Linear System.

❖ Question 1, we have solved it in two methods:

The first method:
You need to write an Augmented Matrix containing the original Matrix and the Identity Matrix (which has the same size as the original Matrix),

Then,
You need to convert the original Matrix into the Identity Matrix using Elementary Row Operations
(Apply Reduced Row Echelon Form (RREF) for the left side of the Augmented Matrix).

The Identity Matrix will convert into the inverse of the original Matrix as long as you apply the same Elementary Row Operations for the Augmented Matrix.

The second method:
You need to do a simple formula to find the inverse of a Matrix.

❖ Question 2, we have used the inverse of a Matrix to solve a Linear System Ax=b. Simply, we do:
x=A^{-1} b.

To illustrate how we do this
Since A^{-1} exists, then, we can multiply both sides of the Linear System Ax=b by A^{-1} from the left.

Then we have
A^{-1} Ax = A^{-1} b

We can simplify this to
I_2 x = A^{-1} b (we use A * A^(-1) = I_2)

We can simplify this to
x = A^{-1} b.

❖ Also, we have explained in this video that if the inverse of the Matrix Does Not Exist (DNE).

Note:
if A inverse exists, then A * A^(-1) = A^(-1) * A = Identity Matrix
If A is not a Square Matrix, then the inverse of Matrix A is DNE.

The Square Matrix is a Matrix with
the number of Rows = the number of Columns.

❖ The number of Rows and Columns that a Matrix has is called its Size, Order, or Dimension.

0:00 ❖ Introduction
1:47 Review how to find the inverse (the steps)
13:30 Solve 1 (The first method)
22:45 Solve 1 (The second method)
27:28 Solve 2
32:57 To confirm the solution x is correct
35:57 Future plan

The link to this playlist (Linear Algebra):
   • Linear Algebra  

My Website:
https://www.Mulkek.com

Subscribe to My Channel to check out more videos:
   / mulkek