Algebra Know-It-ALL

Multiply

We may multiply all the elements in any row by a nonzero constant, keeping the elements in
the same order from left to right. For example, if we have

a 1 b 1 c 1 d 1

a 2 b 2 c 2 d 2

a 3 b 3 c 3 d 3

we can change this to

a 1 b 1 c 1 d 1

ka 2 kb 2 kc 2 kd 2

a 3 b 3 c 3 d 3

In this case, all the elements in the second row have been multiplied by k. Because the absolute
value of k can be smaller than 1, we can extrapolate this rule to let us multiply or divide all the
elements in any row by a nonzero constant. As with the swap move, we can operate only on
entire rows. We can’t do this maneuver with individual elements or with columns.

Add

We may add all the elements in any row to all the elements in another row, and then replace
the elements in either row by the sum, taking care to keep the elements of both rows in the
same order from left to right. Suppose we start with this matrix:

a 1 b 1 c 1 d 1

a 2 b 2 c 2 d 2

a 3 b 3 c 3 d 3

We can change it to either of the following:

a 1 b 1 c 1 d 1

a 1 +a 2 b 1 +b 2 c 1 +c 2 d 1 +d 2

a 3 b 3 c 3 d 3

or

a 1 +a 2 b 1 +b 2 c 1 +c 2 d 1 +d 2

a 2 b 2 c 2 d 2

a 3 b 3 c 3 d 3

Note that the replaced row must be one of the two involved in the sum. In this example, we
aren’t allowed to replace the third row with the sum of the first and second rows.

Matrix Operations 299