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Linear Algebra - Matrices: (lesson 3 of 3)

Inverse of a matrix by Gauss-Jordan elimination

To find the inverse of matrix A, using Gauss-Jordan elimination, we must find a sequence of elementary row operations that reduces A to the identity and then perform the same operations on In to obtain A-1.

Inverse of 2 x 2 matrices

Example 1: Find the inverse of

Solution:

Step 1: Adjoin the identity matrix to the right side of A:

Step 2: Apply row operations to this matrix until the left side is reduced to I. The computations are:

Step 3: Conclusion: The inverse matrix is:

Not invertible matrix

If A is not invertible, then, a zero row will show up on the left side.

Example 2: Find the inverse of

Solution:

Step 1: Adjoin the identity matrix to the right side of A:

Step 2: Apply row operations

Step 3: Conclusion: This matrix is not invertible.

Inverse of 3 x 3 matrices

Example 1: Find the inverse of

Solution:

Step 1: Adjoin the identity matrix to the right side of A:

Step 2: Apply row operations to this matrix until the left side is reduced to I. The computations are:

Step 3: Conclusion: The inverse matrix is: