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4 −3 −−−−−−−−−
Replace R3 with −4R1 + R3
3 The matrix equivalent of ‘triangular form’ is row echelon form. The reader is encouraged to
refer to Deﬁnition 8.3 for comparison. Note that the analog of ‘leading variable’ of an equation
is ‘leading entry’ of a row. Speciﬁcally, the ﬁrst nonzero entry (if it exists) in a row is called the
leading entry of that row.
Definition 8.4. A matrix is said to be in row echelon form provided all of the following
1. The ﬁrst nonzero entry in each row is 1.
2. The leading 1 of a given row must be to the right of the leading 1 of the row above it.
3. Any row of all zeros cannot be placed above a row with nonzero entries.
To solve a system of a linear equations using an augmented matrix, we encode the system into an
augmented matrix and apply Gaussian Elimination to the rows to get the matrix into row-echelon
form. We then decode the matrix and back substitute. The next example illustrates this nicely.
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