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matrix formalism-page3

matrix formalism-page3 - 2 If the top entry of the column...

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Suppose we have an augmented matrix from a linear system. Do the row-operations change the solution set of the linear system? That is, are we sure that, after applying row-operations, we haven’t lost any solutions? It turns out that row-operations do not change the solution set. To prove this, however, we will use invertible matrices and matrix multiplication, so the proof is postponed until later in the semester. Now that we have the terminology, we turn to the systematic method (i.e., Gauss-Jordan elimination) in which we will apply the row -operations to put a matrix A into rref( A ). Example 8. Returning to example 1, we found rref( A ) by applying the Gauss-Jordan elimina- tion algorithm. In this case, it works as follows: In general, Gauss-Jordan elimination proceeds along the following steps, moving from the top row to the bottom row. 1. Find the first nonzero column (starting from the left).
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Unformatted text preview: 2. If the top entry of the column found in step 1 is zero, swap the top row with another row so that the top entry in this column is nonzero. 3. Multiply the top row by a scalar so that the entry in the column from step 1 is 1. 4. Eliminate all other entries in this column by adding a suitable multiple of the top row to the other rows. 5. Move one row down and one column to the right from the row and column that we focused on in the previous steps. If the entry in this new row and new column is zero and all entries below it are also zero then move one column to the right (but keep the same row). Repeat this step as necessary. 6. Return to step 1. The algorithm stops when we run out of rows or columns. Next two examples show how to apply Gauss-Jordan elimination to ±nd the reduced-row-echelon form of a matrix. 3...
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