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The concept of row reduction and elementary row operations for simplifying matrices. Through examples, it explains how to interchange rows, multiply rows by constants, and add or subtract rows to make a matrix row-equivalent. The document also covers the definitions and differences between row echelon form (ref) and reduced row echelon form (rref).
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Now that we know a little bit about matrices, we’re going to learn how to use matrices to solve problems! One of the most useful things we can do to a matrix is to “row reduce” it. Row reduction is a process by which a matrix is simplified into an “equivalent” matrix which is easier to use overall. In order to make this procedure more canonical, we’ll perform our reduction using a very precise collection of operations known as elementary row operations. Throughout, we’ll refer to the matrix
for all of our examples.
There are three elementary row operations that we can perform on a matrix to get a new matrix which is considered “row equivalent” to it:^1
(^1) Definition: Two matrices M and N are said to be row equivalent if there is a series of elementary row operations which transforms M into N (and vice versa).
Unsurprisingly, we can perform these three elementary row operations in succession to provide additional simplification. With a little foresight, this can yield a much simpler matrix which is row-equivalent to the matrix we started with:
Example:
︸ ︷︷ ︸ M
Note that each of the above matrices is row-equivalent to M.
Moving forward, one of our main goals will be to perform these three elementary row operations in succession until we get to a matrix which is in Row Echelon Form (REF) and/or Reduced Row Echelon Form (RREF).