Page 223 - Advanced engineering mathematics

7.3 Reduced Row Echelon Form 203

⎛ ⎞
−3 15 √ 0 −9 14
5. A = ;add 3 times row 2 to row 1, then
2 8 8. A = ⎝ 1 5 2 ⎠ ; interchange rows 2 and 3, then
9 15 0
multiply row 2 by 15, then interchange rows 1 and 2.
add 3 times row 2 to row 3, then interchange rows 1
and 3 and then multiply row 3 by 4.
⎛ ⎞
3 −4 5 9
6. A = ⎝ 2 1 3 −6 ⎠ ; add row 1 to row 3, then In each of Problems 9, 10, and 11, A is an n × m matrix.
1 13 2 6
9. Let B be formed from A by interchanging rows s and
√
add 3 times row 1 to row 2, then multiply row 3 by t.Let E be formed from I n by interchanging these
4,then addrow 2torow3. rows. Prove that B = EA.
10. Let B be formed from A by multiplying row s by α.
⎛ ⎞
−1 0 3 0 Let E be formed from I n by multiplying row s by α.
7. A = ⎝ 1 3 2 9 ⎠ ; multiply row 3 by 4, then Prove that B = EA.
−9 7 −5 7
11. Let B be formed from A by adding α times row s
add 14 times row 1 to row 2 and then interchange rows to row t.Let E be formed from I n by this operation.
3 and 2. Prove that B = EA.

7.3 Reduced Row Echelon Form

Now that we know how to perform elementary row operations, we will address a reason why we
should want to do this. This section establishes a special form that we will want to manipulate
matrices into, and the next two sections apply this special form to the solution of systems of
linear equations.
Define the leading entry of a row of a matrix to be its first nonzero element, reading
from left to right. If all of the elements of a row are zero, then this row has no leading
entry.

An n × m matrix A is in reduced row echelon form if it satisfies the following conditions.

1. The leading entry of each nonzero row is 1.
2. If any row has its leading entry in column j, then all other elements of column j
are zero.
3. If row i is a nonzero row and row k is a zero row, then i < k.
4. If the leading entry of row r 1 is in column c 1 , and the leading entry of row r 2 is in
column c 2 , and r 1 <r 2 , then c 1 < c 2 .

When a matrix satisfies these conditions, we will often shorten “reduced row echelon
form” and simply say that the matrix is reduced,orin reduced form.

Condition (1) of the definition means that, if we look across any nonzero row from left
to right, the first nonzero element we see is 1. Condition (2) means that, if we stand at the
leading entry 1 of any nonzero row and look straight up or down that column, we see only
zeros. Condition (3) means that any row of zeros in a reduced matrix must lie below all rows
having nonzero elements. Zero rows are at the bottom of the matrix. Condition (4) means that
the leading entries of a reduced matrix move downward from left to right as we look at the
matrix.

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 14:23 THM/NEIL Page-203 27410_07_ch07_p187-246