Page 221 - Advanced engineering mathematics

7.2 Elementary Row Operations 201

A 1 = E 1 A
A 2 = E 2 A 1 = E 2 E 1 A
A 3 = E 3 A 2 = E 3 E 2 E 1 A
.
.
.
.
.
.
A r−1 = E r−1 A r−2 = E r−1 E r−2 ···E 2 E 1 A
A r = E r A r−1 = E r E r−1 ···E 2 E 1 A.
If we designate
= E r E r−1 ···E 2 E 1
in this order, then
B = A.
Furthermore is a product of elementary matrices.
We will record this as a theorem.

THEOREM 7.5

Let B be obtained from A by a sequence of elementary row operations. Then there is a matrix
which is a product of elementary matrices such that
B = A.
In forming as a product of elementary matrices, E 1 performs the first row operation on A,
then E 2 performs the second operation on E 1 A, and so on. The order of the operations, hence of
the factors making up , is crucial.
We do not need to actually write down each E j to form . The same result is achieved
as follows: perform the first row operation on I n , then the second operation on the resulting
matrix, then the third operation on this matrix, and so on. After all the row operations have been
performed, the end result is .

EXAMPLE 7.11
Let
⎛ ⎞
0 −11 4
A = 9 3 7 −7 ⎠ .
⎝
0 2 1 5
We will form B by starting with A and performing the following operations in the order given:
O 1 : add −3 times row 2 to row 3; then
O 2 : add 2 times row 1 to row 2; then
O 3 : interchange rows 1 and 3; then
O 4 : multiply row 2 by −4.
To form to perform these operations, begin
⎛ ⎞ ⎛ ⎞
1 0 0 1 0 0
O 1 O 2
I 3 −→ 0 1 0 ⎠ −→ 2 1 0 ⎠
⎝
⎝
0 −31 0 −31

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October 14, 2010 14:23 THM/NEIL Page-201 27410_07_ch07_p187-246