❖ In this video, we have explained how to solve a linear system using the inverse of a matrix, which is a precalculus video tutorial. This video builds on our previous tutorials where we demonstrated how to find the inverse of 2x2, 3x3, and 4x4 matrices.

❖ To find the inverse of a matrix, you need to create an augmented matrix containing the original matrix and the identity matrix (which has the same size as the original matrix).

Then,
You need to transform the original matrix into the identity matrix using Elementary Row Operations
(means apply Reduced Row Echelon Form (RREF) to the left side of the augmented matrix).

By applying the same Elementary Row Operations to the augmented matrix for the right side, the identity matrix (which is on the right side of the augmented matrix) will transform into the inverse of the original matrix.

So,
The identity matrix (which is on the left side of the augmented matrix) will convert into the inverse of the original matrix as long as you apply the same Elementary Row Operations for the augmented matrix.

❖ To confirm your answer, multiply the original matrix by its inverse. If the product is the identity matrix, then you have found the correct inverse.

This can be represented as:
(The matrix A) multiplied by (the inverse of the matrix A) equal to (the identity matrix) of the same size as A,

so, if the inverse of matrix A exists then
A * A^(-1) = Identity matrix.
where A is the original matrix, and A^(-1) is its inverse.

❖ In this video, we also explain when the inverse of a matrix does not exist (DNE).

Note the following cases:
If A is not a Square Matrix, then the inverse of matrix A is DNE.
If the linear system Ax=b has no solution, then the inverse of matrix A is DNE.
If the linear system Ax=b has infinitely many solutions, 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 & start solving Example 1
5:49 Matrix inverse definition: understand the concept
8:35 Review how to find the inverse (the steps)
18:35 Important notes
21:42 Examples for how to use A^(-1)
27:09 Continue to solve Example 1
40:12 Confirm that A*A^(-1) = identity
54:23 Confirm that our answer x is correct
57:38 Solve Example 2
1:13:15 Conclusion

By watching this tutorial, you will gain a deeper understanding of matrix inverses and how to apply them in solving linear systems.

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

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