Solve By rref(

Return to Roadmap 2

We will solve Example 12 from the Chapter 2 Roadmap page by using the built-in function rref(. Start with the initial system of four linear equations in four variables.

-5x₁ + 7x₂ + 9x₃ - 11x₄	=	56	(1)
1x₁ + 2x₂ - 6x₃ - 3x₄	=	115	(2)
12x₁ - 4x₂ + 3x₃ + 8x₄	=	- 74	(3)
5x₁ - 7x₂ - 1x₃ + 4x₄	=	- 73	(4)

Once we make sure that we have accounted for all of the variables in all of the equations and that the variables are in the same order in all equations, we can represent that system of linear equations via the matrix

As noted in an earlier page we could solve this system of four linear equations in four variables by methodically applying elementary row operations. The methodical part of that really comes down to first working down the main diagonal of values, changing each diagonal to the value 1 and then creating 0's below the diagonal element of the matrix. Then, the second part of the process works up that main diagonal creating 0's above the diagonal elements. The result, in the case of our set of equations is

That form is called reduced row–echelon form. Because the process to produce this form is so methodical, it can be captured in a program. The TI-83/84 family of calculators refer to this function as rref(. We will use that function to solve the problem.

Figure 1	First we need to have the matrix defined in the calculator. Earlier pages have demonstrated doing this. Here we just verify, in Figure 1, that the matrix is appropriately defined as [A].
Figure 2	We find rref( in the MATH tab of the MATRIX window. Remember that we get there via the key sequence, followed by using the key to move the highlight to the MATH tab. The we use the key to move the selection highlight down to the rref( option, as shown in Figure 2. Once there, we can press the key to paste the function name onto our main screen as is shown in Figure 3.
Figure 3	To complete the command we need to supply the name of the matrix we are using. As we have seen in earlier pages, to input the name of our matrix we need to return to the MATRIX window via the keys.
Figure 4	Figure 4 shows the MATRIX window on the calculator used for these images. We know, from Figure 1, that the desired matrix is [A]. That is the currently selected matrix so we just need to press the key to paste that onto the main screen.
Figure 5	Then we need to complete the command with a closing parenthesis, , as shown in Figure 6.
Figure 6	Once the command is formed, as it is here in Figure 6, we just press to get the calculator to perform the function.
Figure 7	Figure 7 shows the result. That one statement, rref([A]), did all of the work that we had to go through in using the elementary row operations to generate the reduced row-echelon form of the matrix. And, from this form we can read out the solution, x₁ = 6, x₂ = 11, x₃ = ^–10, and x₄ = ^–9.