Gaussian elimination method to find inverse of matrix
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Example (3 × 3)
Find the inverse of the matrix A using Gauss-Jordan elimination.
A = 41314 129715510
Our Procedure
We write matrix A on the left and the Identity matrix I on its right separated with a dotted line, as follows. The result is called an augmentedmatrix.
We include row numbers to make it clearer.
41314129715510100 Row[1]010Row[2]001Row[3]
Next we do several row operations on the 2 matrices and our aim is to end up with the identity matrix on the left, like this:
100010001??? Row[1]???Row[2]???Row[3]
(Technically, we are reducing matrix A to reduced row echelon form, also called row canonical form).
The resulting matrix on the right will be the inverse matrix of A.
Our row operations procedure is as follows:
We get a "1" in the top left corner by dividing the first rowThen we get "0" in the rest of the first columnThen we need to get "1" in the second row, second columnThen we make all the other entries in the second column "0".
We keep going like this until we are left with the identity matrix on the left.
Let's now go ahead and find the inverse.
Solution
We start with:
41314129715510100 Row[1]010Row[2]001Row[3]
Find the inverse of the matrix A using Gauss-Jordan elimination.
A = 41314 129715510
Our Procedure
We write matrix A on the left and the Identity matrix I on its right separated with a dotted line, as follows. The result is called an augmentedmatrix.
We include row numbers to make it clearer.
41314129715510100 Row[1]010Row[2]001Row[3]
Next we do several row operations on the 2 matrices and our aim is to end up with the identity matrix on the left, like this:
100010001??? Row[1]???Row[2]???Row[3]
(Technically, we are reducing matrix A to reduced row echelon form, also called row canonical form).
The resulting matrix on the right will be the inverse matrix of A.
Our row operations procedure is as follows:
We get a "1" in the top left corner by dividing the first rowThen we get "0" in the rest of the first columnThen we need to get "1" in the second row, second columnThen we make all the other entries in the second column "0".
We keep going like this until we are left with the identity matrix on the left.
Let's now go ahead and find the inverse.
Solution
We start with:
41314129715510100 Row[1]010Row[2]001Row[3]
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