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Inverse of a matrix by Gauss-Jordan elimination
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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:
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