Algebra Know-It-ALL

302 The Matrix Morphing Game

Formatting the equations

None of these equations is in the proper form for assembling a matrix. Let’s get them that

way! Fortunately, the maneuvers are simple. With the first equation, we can subtract 2y from

each side to get

3 x− 2 y+z= 11

With the second equation, we can subtract x from each side to obtain

−x+ 4 y+ 2 z= 0

With the third equation, we can subtract 3z from each side, getting

− 5 x+y− 3 z=− 20

We now have the equations in form, and can state the whole system like this:

3 x− 2 y+z= 11

−x+ 4 y+ 2 z= 0

− 5 x+y− 3 z=− 20

Building the matrix

To construct the matrix, we remove all the variables and arrange the remaining numbers into

an orderly array, paying close attention to the signs:

3 − 2 111

− 1 420

− 5 1 − 3 − 20

Deriving the echelon form

There are many different routes by which we can arrive at an echelon form of this matrix.

Let’s start by getting 0 at the extreme left in the bottom row. We can multiply the second row

by −5 to obtain

(^3) − 2 111
(^5) − 20 − 10 0
− 5 1 − 3 − 20
We can add the second and third rows and then replace the third row with the sum to get
(^3) − 2 111
5 − 20 − 10 0
0 − 19 − 13 − 20