MATH 110: Course Materials - operations on a matrix

COLLEGE ALGEBRA MATH 110		Southwestern Adventist University
Distance Education	Lawrence E. Turner, Jr., Ph.D.

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Operations on a Matrix

If we have a set of simultaneous linear equations, there are certain operations that we can perform on the set that do not change the value of the solution set. Recasting these equations in matrix form is actually only a short-hand notation that makes it easier to solve the system without having to write and rewrite all the variable names at each step.

It should be no surprise that the operations on the rows of the matrix come directly from the things we can do with the original set of equations.

As an example, consider the set of three equations in three unknowns:

2x − 3y − 4z = 0

x + 2y + z = 4

3x + y + 3z = 2

We want operations that we can perform on this, or any set of equations, that will not change the results; ie. the set of values of the variables that will satisfy this set of equations—the solution set.

Changing the order in which the equations are written will leave the solution set the same. Therefore, our first operation is:

1. We can exchange any two equations.

Since these are three independent equations, it does not matter in which order we write them.

Exchanging the first two equations results in:

x + 2y + z = 4

2x − 3y − 4z = 0

3x + y + 3z = 2

This set of equations has the same solution set as our original set.

The second operation comes from the property of equations that we can multiply (or divide) both sides by the same non-zero value, and the result does not change. Therefore, our second operation is:

2. We can multiply (or divide) both sides of any equation by the same non-zero value.

As an example, if we multiply the original second equation by 2, then we get:

2x − 3y − 4z = 0

2x + 4y + 2z = 8

3x + y + 3z = 2

The third operation comes from the property of equations that we can add to (or subtract from) both sides of an equation the same value and the result does not change. However, to get it in a useful form requires some elaboration.

A simple application of adding the value 4 to both sides of the original third equation would give us:

2x − 3y − 4z = 0

x + 2y + z = 4

3x + y + 3z + 4 = 6

While this is true, it is not helpful in this form. However, we observe that according to the original second equation:

x + 2y + z = 4

This states that the left-hand side has the value 4. Therefore, let us add the left-hand side with a value of 4 to the left-hand side of the third equation and add the numeric value 4 to the right-hand side. The important thing is we are adding the same value to both sides even though the two sides are in a different form. Performing this, we get:

2x − 3y − 4z = 0

x + 2y + z = 4

4x + 3y + 4z = 6

Actually, we could have multiplied both sides of the second equation by any non-zero value before adding left-hand side to left-hand side and right-hand-side to right-hand side, and we would not have changed the values of the solution set. As an example, let us add 2 times the second equation to the third; that is, add the value 8 to the third equation:

2x − 3y − 4z = 0

x + 2y + z = 4

5x + 5y + 5z = 10

With the understanding that we add left-hand side to left-hand side and right-hand side to right-hand side, we get our third operation:

3. We can add a non-zero multiple of any equation to any other equation.

To simplify things we write the equations as an augmented matrix. This is simply an array of numbers with the left part of each row taken from the coefficients of the variables and the right-most element is the constant left-hand side. The augmented matrix has one more column than a matrix formed just from the coefficients of the variables. For the set of equations above, the augmented matrix becomes:

The operations that we can perform on this array come directly from the three operations that we can perform on a set of simultaneous linear equations:

Any two rows can be exchanged.

Any row may be multiplied (or divided) by a non-zero constant.

A multiple of any row can be added to any other row.

We can diagram these as:

1.	R_i ⇔ R_k	exchange row i and k
2.	c·R_i → R_i	multiply row i by the value c to get the new row i
3.	c·R_i + R_k → R_k	add a multiple (c times) of row i to row k to get the new row k

Applying any of these operations to the augmented matrix will not change the solution set just like applying the analogous operations on the set of equations does not change the answer.

Please note carefully what each of these operations do. In particular, if we multiply row 2 by a 7 and add it to row 3, then only row 3 is changed−row 2 remains in its original state. The last two operations, of our set of three, change only the "target" row. This Rule is not something like: replace row 3 with 4 times row 3 plus row 1; that is, we do not multiply the target row by anything—we only add something to it.