Question

Find the inverse of a matrix using Gauss-Jordan method (or by (4*) if n=2).

Answer 1

To find the inverse of a matrix using the Gauss-Jordan method, start with the original matrix and an identity matrix of the same size. Perform row operations to transform the original matrix into the identity matrix, keeping track of the operations performed on the identity matrix as well. If the original matrix can be transformed into the identity matrix, the corresponding operations performed on the identity matrix will give you the inverse of the original matrix.

Step-by-step explanation:

To find the inverse of a matrix using the Gauss-Jordan method, follow these steps:

1. Start with the original matrix and an identity matrix of the same size.
2. Perform row operations to transform the original matrix into the identity matrix, keeping track of the operations performed on the identity matrix as well.
3. If the original matrix can be transformed into the identity matrix, the corresponding operations performed on the identity matrix will give you the inverse of the original matrix.

For example, let's say we have a 2x2 matrix A:

A = |a b| = |1 2|

We can create an identity matrix of the same size:

I = |1 0|

Now, we perform row operations on the matrices:

[A I] = |1 2 1 0|

By performing appropriate row operations:

[A I] = |1 0 -1 2|

Now we have the identity matrix and the inverse of A:

A^-1 = |-1 2|