Part 7 : Inverses and Gauss-Jordan Elimination
Inverse of a matrix
Assuming a matrix A.
The inverse of matrix A will be another matrix of same order, which on multiplication with A return I (Identity Matrix) as product.
Inverse of A is denoted with letter representing matrix raised to power (-1).
Multiplication of matrices is not commutative but when we multiply a matrix with its inverse multiplication can be commutative.
Also, only square matrices have inverses.
Division of matrices was not discussed in part 2.
Now, with the help of inverses we can perform division of two matrices.
Taking an example
This is equivalent to multiplication of matrix B with inverse of A.
An algorithm to find inverse of a given matrix, it is similar to Gaussian elimination or we can say it is Gaussian elimination extended to one more step.
Wilhelm Jordan improved the stability of algorithm.
To perform Gauss-Jordan Elimination we have to :
1. Make augmented matrix from given matrix and its identity matrix (Order of Identity matrix is decided according to the order of given matrix).
2. Perform elimination (as in step 2 of Gaussian elimination), aiming to obtain row echelon form on left half of augmented matrix.
3. Reduce it further to get Reduced Row Echelon Form (Identity matrix) on left half of augmented matrix.
4.The right half of augmented matrix, is the inverse of given matrix.
Assuming that we have to find inverse of matrix A (above) through Gauss-Jordan Elimination.
Step 1 (Make Augmented matrix) :
Step 2 (Elimination) :
Applying the operations
Now, we apply similar operations for column 2.
If we were doing Gaussian Elimination, we would’ve finished by now and started back substitution but to find inverse we have to take it one step further. We have to find Reduced Row Echelon Form
Step 3 (Elimination to Reduced Row Echelon Form) :
For obtaining reduced row echelon form, we aim to convert left half of augmented matrix into an identity matrix, by using similar operations as in Gaussian Elimination and we start from bottom row to top.
First, we reduce row 3 to an identity matrix form.
Then, we reduce row 2.
Now, we reduce row 1 and we will get reduced row echelon form.
Step 4 (Separate right half of augmented matrix) :
The right half of augmented matrix is the inverse of matrix A.
Are all matrices invertible?
No, a matrix should satisfy the following two conditions to be invertible.
1. It should be a square matrix.
2. It should not be a singular matrix (its determinant is not equal to 0).
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