Algebra Know-It-ALL

298 The Matrix Morphing Game

Now we have these three equations that make up the linear system:

0 x+ 7 y− 3 z= 3

2 x+ 0 y+ 8 z=− 7

12 x− 7 y+ 0 z= 0

We may want to write the above equations like this, so we are sure to get the signs of the coefficients right:

0 x+ 7 y+ (− 3 z)= 3

2 x+ 0 y+ 8 z=− 7

12 x+ (− 7 y)+ 0 z= 0

We can write this system in matrix form by removing the variables and equals signs, and then aligning the

coefficients into neat rows and columns:

07 − 3 3

(^208) − 7
12 − 7 00

Matrix Operations

Imagine the matrix for a three-by-three linear system as a game board with 12 positions,

arranged in three horizontal rows and four vertical columns. Let’s invent a matrix morphing

game. There are three types of moves in this game: swap,multiply, and add. We can make as

many of these moves as we want.

Swap

We may interchange all the elements between two rows in a matrix, while keeping the elements

of both rows in the same order from left to right. For example, if we start with

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

a 3 b 3 c 3 d 3

a 2 b 2 c 2 d 2

a 1 b 1 c 1 d 1

In this case, the first and third rows have been swapped. Note that we cannot swap individual

elements or vertical columns! The swap maneuver is only allowed between entire rows.