Put the following three-by-three linear system into matrix form: 7y = 3z + 3 8z = 2x 7 12x = 7y

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None of these three equations is in the proper form for conversion to matrix notation We ll have to manipulate them Here are the processes, step-by-step For the first equation: 7y = 3z + 3 0x + 7y = 3z + 3 0x + 7y 3z = 3 For the second equation: 8z = 2x 7 2x + 8z = 7 2x + 0y + 8z = 7 For the third equation: 12x = 7y 12x 7y = 0 12x 7y + 0z = 0

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298 The Matrix Morphing Game

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Now we have these three equations that make up the linear system: 0x + 7y 3z = 3 2x + 0y + 8z = 7 12x 7y + 0z = 0 We may want to write the above equations like this, so we are sure to get the signs of the coefficients right: 0x + 7y + ( 3z) = 3 2x + 0y + 8z = 7 12x + ( 7y) + 0z = 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: 0 2 12 7 0 7 3 8 0 3 7 0

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Matrix Operations

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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

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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

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a1 a2 a3 we can change it to a3 a2 a1 b3 b2 b1 c3 c2 c1 d3 d2 d1 b1 b2 b3 c1 c2 c3 d1 d2 d3

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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

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Matrix Operations

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

a1 a2 a3 we can change this to a1 ka2 a3 b1 kb2 b3 c1 kc2 c3 d1 kd2 d3 b1 b2 b3 c1 c2 c3 d1 d2 d3

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:

a1 a2 a3 b1 b2 b3 c1 c2 c3 d1 d2 d3

We can change it to either of the following:

a1 a1 + a2 a3 or a1 + a2 a2 a3 b1 + b2 b2 b3 c1 + c2 c2 c3 d1 + d2 d2 d3 b1 b1 + b2 b3 c1 c1 + c2 c3 d1 d1 + d2 d3

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

300 The Matrix Morphing Game

The final goal The matrix morphing game, like any sensible game, has an ultimate objective Our goal is to get a matrix representing a three-by-three linear system into this form:

1 0 0 0 1 0 0 0 1 x y z

where x, y, and z are real numbers This is called the unit diagonal form Now imagine that we start with a three-by-three linear system, make it into a matrix, and then play the matrix morphing game until we get the unit diagonal form Do you suspect that the values x, y, and z, which appear in the far right column, will represent the solution to the linear system If so, you are right, provided the system is consistent (has a unique solution)