Page 219 - Advanced engineering mathematics

7.2 Elementary Row Operations 199

EXAMPLE 7.9
We will look at an example of each of these row operations. Let
⎛ ⎞
−2 1 6 −3
1 1 2 5
⎜ ⎟
A = ⎜ ⎟ .
⎝ 0 9 3 −7 ⎠
2 −34 11
If we interchange rows two and three of A, we obtain
⎛ ⎞
−2 1 6 −3
0 9 3 −7
⎜ ⎟
⎜ ⎟ .
⎝ 1 1 2 5 ⎠
2 −34 11
If we multiply row three of A by 7, we obtain
⎛ ⎞
−2 1 6 −3
⎜ 1 1 2 5 ⎟
⎜ ⎟ .
⎝ 0 63 21 −49 ⎠
2 −3 4 11
And if we add −6 times row one to row three of A, we obtain
⎛ ⎞
−2 1 6 −3
⎜ 1 1 2 5 ⎟
⎜ ⎟ .
⎝ 12 3 −33 11 ⎠
2 −3 4 11
Every elementary row operation can be performed by multiplying A on the left by a square
matrix constructed by applying that row operation to an identity matrix.

THEOREM 7.4

Let A be an n × m matrix. Suppose B is formed from A by an elementary row operation. Let E
be the matrix formed by performing this row operation on I n . Then
B = EA.

A matrix formed by performing an elementary row operation on I n is called an elementary
matrix. Theorem 7.4 says that we can perform any elementary row operation on A by multiplying
A on the left by the elementary matrix formed by performing this row operation on I n .Weleave
a proof of this to Exercises 7.9, 7.10, and 7.11. However, it is instructive to see the theorem in
action.

EXAMPLE 7.10
Let
⎛ ⎞
−216 −3
⎝ 1
A = 1 2 5 ⎠ .
0 9 3 −7
Since A is 3 × 4, we will use I 3 to perform row operations.

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October 14, 2010 14:23 THM/NEIL Page-199 27410_07_ch07_p187-246