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7.3 Reduced Row Echelon Form 207

EXAMPLE 7.15
Let

−3 1 0
A = .
4 −2 1
We will reduce A and, at the same time, find a 2 × 2matrix such that A = A R . Start with the
augmented matrix and reduce it:
⎛ ⎞
.
. . −3 1 0 . . 1 0
[A.I 2 ]= ⎝ ⎠
.
4 −21 . . 0 1
⎛ ⎞
.
1 −1/3 0 . . −1/3 0
((−1/3) times row one ) → ⎝ ⎠
.
4 −2 1 . . 0 1
.
⎛ ⎞
1 −1/3 0 . . −1/3 0
(add − 4 times row one to row two ) → ⎝ ⎠
.
0 −2/3 1 . . 4/3 1
.
⎛ ⎞
1 −1/3 0 . . −1/3 0
(multiply row two by − 3/2) → ⎝ ⎠
.
0 1 −3/2 . . −2 −3/2
.
⎛ ⎞
1 0 −1/2 . . −1 −1/2
( add 1/3rowtwotorowone ) → ⎝ ⎠
.
0 1 −3/2 . . −2 −3/2
.
.
=[A.I 2 ] R .
The first three columns of this reduced augmented matrix are A R , while the last two columns
form :

1 0 −1/2 −1 −1/2
A R = and = .
0 1 −3/2 −2 −3/2
As a check,

−1 −1/2 −3 1 0 1 0 −1/2
A = = = A R .
−2 −3/2 4 −21 0 1 −3/2
.
.
This is the reduced form of [A.I 2 ].
MAPLE’s pivot command is well suited to reducing a matrix A which has been entered
into the program. First look for the leading entries of the nonzero rows. The location of a leading
entry is called a pivot position. We obtain zeros above and below a leading entry by elementary
row operations, adding constant multiples of this row to the other rows if necessary. This is called
pivoting about this leading entry, and can be done in one operation which in MAPLE is called
pivot. If a leading entry α occurs in the i, j position of A, we can form a matrix B having zeros
above and below α by entering
B := pivot(A,i,j);

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October 14, 2010 14:23 THM/NEIL Page-207 27410_07_ch07_p187-246