5-30-2021 Row Reduction Row reduction (or Gaussian elimination) is the process of using row operations to reduce a matrix to row reduced echelon form. This procedure is used to solve systems of linear equations, invert matrices, compute determinants, and do many other things. I’ll begin by describing row operations, after which I’ll show how they’re used to do a row reduction. Warning: Row reduction uses row operations. There are similar operations for columns which can be used in other situations (like computing determinants), but not here. There are three kinds of row operations. (Actually, there is some redundancy here — you can get away with two of them.) In the examples below, assume that we’re using matrices with real entries (but see the notes under (b)). (a) You may swap two rows. Here is a swap of rows 2 and 3. I’ll denote it by r2 r3. ↔ 1 2 3 1 2 3 4 5 6 7 8 9 → 7 8 9 4 5 6 (b) You may multiply (or divide) a row by a number that has a multiplicative inverse. If your number system is a field, a number has a multiplicative inverse if and only if it’s nonzero. This is the case for most of our work in linear algebra. If your number system is more general (e.g. a commutative ring with identity), it requires more care to ensure that a number has a multiplicative inverse. Here is row 2 multiplied by π. I’ll denote this operation by r2 πr2. → 1 2 3 1 2 3 4 5 6 4π 5π 6π → 7 8 9 7 8 9 (c) You may add a multiple of a row to another row. For example, I’ll add 2 times row 1 to row 2. Notation: r2 r2 +2r1. → 1 1 5 1 1 5 2− 4 2 0− 2 12 − → 1 3 1 1 3 1 Notice in “r2 r2 +2r1” row 2 is the row that changes; row 1 is unchanged. You figure the “2r1” on scratch paper, but→ you don’t actually change row 1. You can do the arithmetic for these operations in your head, but use scratch paper if you need to. Here’s the arithmetic for the last operation: r2 = 2 4 2 − 2r1 = 2 2 10 − r2 +2r1 = 0 2 12 Since the “multiple” can be negative, you may also subtract a multiple of a row from another row. 1 For example, I’ll subtract 4 times row 1 from row 2. Notation: r2 r2 4r1. → − 1 2 3 1 2 3 4 5 6 0 3 6 → − − 7 8 9 7 8 9 Notice that row 1 was not affected by this operation. Likewise, if you do r17 r17 56r31, row 17 changes and row 31 does not. → − Operation (c) is probably the one that you will use the most. You may wonder: Why are these operations allowed, but not others? We’ll see that these row operations imitate operations we can perform when we’re solving a system of linear equations. Example. In each case, tell whether the operation is a valid single row operation on a matrix with real numbers. If it is, say what it does (in words). (a) r5 r3. ↔ (b) r6 r6 7. → − (c) r3 r3 + πr17. → (d) r6 5r6 + 11r2. → (e) r3 r3 + r4 and r4 r4 + r3. → → (a) This operation swaps row 5 and row 3. (b) This isn’t a valid row operation. You can’t add or subtract a number from the elements in a row. (c) This adds π times row 17 to row 3 (and replaces row 3 with the result). Row 17 is not changed. (d) This isn’t a valid row operation, though you could accomplish it using two row operations: First, multiply row 6 by 5; next, add 11 times row 2 to the new row 6. (e) This is not a valid row operation. It’s actually two row operations, not one. The only row operation that changes two rows at once is swapping two rows. Matrices can be used to represent systems of linear equations. Row operations are intended to mimic the algebraic operations you use to solve a system. Row-reduced echelon form corresponds to the “solved form” of a system. A matrix is in row reduced echelon form if the following conditions are satisfied: (a) The first nonzero element in each row (if any) is a “1” (a leading entry). (b) Each leading entry is the only nonzero element in its column. (c) All the all-zero rows (if any) are at the bottom of the matrix. (d) The leading entries form a “stairstep pattern” from northwest to southeast: 01600 2 ... 00010 1 ... − 00001 4 ... 00000 0 ... . . In this matrix, the leading entries are in positions (1, 2), (2, 4), (3, 5), ... 2 Here are some matrices in row reduced echelon form. This is the 3 3 identity matrix: × 1 0 0 0 1 0 0 0 1 The leading entries are in the (1, 1), (2, 2), and (3, 3) positions. This matrix is in row reduced echelon form; its leading entries are in the (1, 1) and (2, 3) positions. 1 5 02 0 0 13 0 0 00 A leading entry must be the only nonzero number in its column. In this case, the “5” in the (1, 2) position does not violate the definition, because it is not in the same column as a leading entry. Likewise for the “2” and “3” in the fourth column. This matrix has more rows than columns. It is in row reduced echelon form. 1 0 0 1 0 0 0 0 This row reduced echelon matrix has leading entries in the (1, 1), (2, 2), and (3, 4) positions. 1060 2 0 1 1 0 3 − 0001 4 0000 0 The nonzero numbers in the third and fifth columns don’t violate the definition, because they aren’t in the same column as a leading entry. A zero matrix is in row-reduced echelon form, though it won’t normally come up during a row reduc- tion. 0 0 00 0 0 00 0 0 00 Note that conditions (a), (b), and (d) of the definition are vacuously satisfied, since there are no leading entries. Just as you may have wondered why only certain operations are allowed as row operations, you might wonder what row reduced echelon form is for. “Why do we want a matrix to look this way?” As with the question about row operations, the rationale for row reduced echelon form involves solving systems of linear equations. If you want, you can jump ahead to the section on solving systems of linear equations and see how these questions are answered. In the rest of this section, I’ll focus on the process on row reducing a matrix and leave the reasons for later. Example. The following real number matrices are not in row reduced echelon form. In each case, explain why. 1 0 0 (a) 0 7 0 0 0 1 1 0 3 (b) 01− 5 . 00 1 3 000 0 (c) 0 1 0 1 . − 001 9 1 37 2 1 (d) 0 1 3− 0 . − 00 0 0 0 1 7 10 (e) 1 0 4 5 . − 000 0 (a) The first nonzero element in row 2 is a “7”, rather than a “1”. (b) The leading entry in row 3 is not the only nonzero element in its column. (c) There is an all-zero row above a nonzero row. (d) The leading entry in row 2 is not the only nonzero element in its column. (e) The leading entries do not form a “stairstep pattern” from northwest to southeast. Row reduction is the process of using row operations to transform a matrix into a row reduced echelon matrix. As the algorithm proceeds, you move in stairstep fashion from “northwest” to “southeast” through different positions in the matrix. In the description below, when I say that the current position is (i, j), I mean that your current location is in row i and column j. The current position refers to a location, not the element at that location (which I’ll sometimes call the current element or current entry). The current row means the row of the matrix containing the current position and the current column means the column of the matrix containing the current position. Some notes: 1. There are many ways to arrange the algorithm. For instance, another approach gives the LU- decomposition of a matrix. 2. Trying to learn to row reduce by following the steps below is pretty tedious, and most people will want to learn by doing examples. The steps are there so that, as you’re learning to do this, you have some idea of what to do if you get stuck. Skim the steps first, then move on to the examples; go back to the steps if you get stuck. 3. As you gain experience, you may notice shortcuts you can take which don’t follow the steps below. But you can get very confused if you focus on shortcuts before you’ve really absorbed the sense of the algorithm. I think it’s better to learn to use a correct algorithm “by the book” first, the test being whether you can reliably and accurately row reduce a matrix. Then you can consider using shortcuts. 4. The algorithm is set up so that if you stop in the middle of a row reduction — maybe you want to take a break to have lunch — and forget where you were, you can restart the algorithm from the very start.