Given a system of linear equations, we first express it as an augmented matrix. We work on the augmented matrix by performing a sequence of the so-called [b]elementary row operations[/b] - the operations corresponding to how we manipulate the equations in the system during the variable-eliminating process. There are only three elementary row operations:[br][br][list][*]([b]Interchange[/b]) Interchange two rows ([u]Notation[/u]: [math]R_i\leftrightarrow R_j[/math] means interchange i[sup]th[/sup] and j[sup]th[/sup] row)[/*][*]([b]Scaling[/b]) Multiply all entries in a row by a nonzero constant ([u]Notation[/u]: [math]kR_i\rightarrow R_i[/math] means multiply i[sup]th[/sup] row by [math]k[/math])[/*][*]([b]Replacement[/b]) Replace one row by the sum of itself and a multiple of another row ([u]Notation[/u]: [math]R_i+kR_j\rightarrow R_i[/math] means replace i[sup]th[/sup] row by the sum of i[sup]th[/sup] row and [math]k[/math] times j[sup]th[/sup] row)[br][/*][/list][br]Notice that these three elementary row operations are [u]reversible[/u] - [math]R_i\leftrightarrow R_j[/math] can reverse [math]R_i\leftrightarrow R_j[/math], [math]\frac 1k R_i\rightarrow R_i[/math] can reverse [math]kR_i\rightarrow R_i[/math] and [math]R_i+(-k)R_j\rightarrow R_i[/math] can reverse [math]R_i+kR_j\rightarrow R_i[/math].[br][br]In fact, elementary row operations can be applied to any matrix, not just an augmented matrix. Two matrices are called [b]row equivalent[/b] if there is a sequence of elementary row operations that transforms from one to another. The following is the important result that we will prove later in this chapter:[br][br][u]Theorem[/u]: If the augmented matrices of two linear systems are row equivalent, then the two systems have the same set of solutions.[br][br]Gaussian elimination is the procedure of transforming the augmented matrix of a linear system by a sequence of elementary row operations to the one in the so-called [b]echelon form[/b]. And you will see that it is easy to obtain the solution(s) to the linear system with its augmented matrix in echelon form and hence solve the original linear system.[br][br]You can try the "[url=http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi?c=roc]Row operation calculator[/url]" in "Linear Algebra Toolkit" developed by P. Bogacki.[br][br]

Given an augmented matrix of a system of linear equation, we say that a row in the matrix a [b]nonzero row[/b] if at least one its entries is a nonzero entry. The [b]leading entry[/b] is the leftmost nonzero entry (in a nonzero row).[br][br]A matrix is in [b]echelon form[/b] (or [b]row echelon form[/b]) if it has the following three properties:[br][list=1][*]All nonzero rows are above any rows of all zeros.[/*][*]Each leading entry of a row is in a column to the right of the leading entry of the row above it.[/*][*]All entries in a column below a leading entry are zeros.[/*][/list][br]If a matrix in echelon form satisfies the following additional conditions, then it is in reduced echelon form (or reduced row echelon form):[br][br][list][*]The leading entry in each nonzero row is 1.[/*][*]Each leading 1 is the only nonzero entry in its column.  [br][/*][/list]

An augmented matrix may be [b]row reduced[/b] (that is, transformed by a sequence of elementary row operations) to more than one matrix in echelon form. However, it can be shown that any matrix is certainly row reduced to a unique matrix in reduced echelon form.[br][br]In each of the following questions, you need to determine whether an augmented matrix is in (reduced) echelon form or not:[br]