7b- Practice problems for Gauss Jordan elimination.

Last Updated on January 11, 2026 by Maged kamel

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Practice problems for Gauss Jordan elimination.

Practice problems for Gauss Jordan elimination.

Practice problems for Gauss Jordan elimination-first practice problem.

We will introduce a practice problem Gauss Jordan elimination, or the reduced echelon form. Practice problem number #33 is from Stewart’s book College Algebra.

For the first practice problem #33 for Gauss-Jordan elimination. Solve the following systems of linear equations. We have three simultaneous equations as: x+2y-Z=-2& x+z=0 and 2*x-y-Z=-3.

Practice problems for Gauss Jordan elimination a11 as a pivot to eliminate a21 and a31.

We will write these equations in augmented form for the given linear equation system.

We will introduce a vertical line between the third and fourth columns to show the augmented matrix. The next step is to set a21, or the second row/first column, to zero.

Make sure that the element a11 is equal to 1, then add (-a21/a11)*R1 to row 2 to get the zero value. In this process, we consider a11 as the pivot element. In our practice problem #1, we will multiply R1 by (-1) and add the result to the second row.

Similarly, let a31 or the third row/first column equal zero. Add (-a31/a11)*R1 to row 3 to get the zero value. Again, in this process, we consider a11 as the pivot element. We will multiply R1 by (-2) and add the result to the third row. Please refer to the following slide image for more details.

Practice problem-1 for Gauss Jordan elimination

Carry on the row operations. Make sure that element a22 is equal to 1, and this is done by multiplying row 2 by (-1/2).

Use a22 as a pivot to eliminate a32.

Use a22 as a pivot by adding R3 to -(a23/a21)*R2. Perform the row operations to obtain the reduced echelon form, or simply an upper triangular matrix with three ones on the diagonal.

We will multiply R2 by (+5) and add the result to the third row. Please refer to the following slide image for more details.

use a11 as a pivot and eliminate both a21 and a31.

Refer to the slide image to see the necessary row operations for Gauss elimination in reduced row echelon form. We divide the third row by -4, and we get an upper matrix with diagonal elements equal to one.

create an upper matrix by dividing R3 by -4.

Use a33 as a pivot to eliminate a23& a13.

We will multiply R3 by (1) and add the result to the first row. We will multiply R3 by (1) and add the result to the second row. Please refer to the following slide image for more details.

Eliminate both a23 and a13 elements by using a33 as a pivot.

Use a22 as a pivot to eliminate a12.

We will use the element a22 to eliminate a12; this is done by adding(-a12/a22)*R2 to R1. Now we have obtained the required reduced row echelon form, or Gauss-Jordan; please refer to the next slide for clarity.

Use a22 as a pivot and let a12 equals zero .

We can then get the values of Z=1&Y=0 and X=-1. In this method, we do not need to make back substitutions. However, it is important to check the Values of X, Y, and Z in any of the three systems of equations to ensure that the solution is accurate.

Check the values of x-y and z by substitution for problem-38

Practice problems for Gauss Jordan elimination-second practice problem.

1-For the second practice problem #38 for Gauss-Jordan elimination. Solve the following systems of linear equations. Solve the following systems of linear equations. We have three simultaneous equations as: 10x+10y-20Z=60& 15x+20y+30z+=-25 and -5*x+30y-10Z=45. The following slide image shows the details of the matrix given for practice problem #38.

We will write these equations in augmented form for the given linear equation system.

We will introduce a vertical line between the third and fourth columns to show the augmented matrix. The next step is to let a21 or the second row/first column be equal to zero.

Make sure element a11 equals 1. First, we will divide the first row by 10. We will add (-a21/a11)*R1 to row 2 to get the zero value. In this process, we consider a11 as the pivot element.

Similarly, let a31 or the third row/first column equal zero. Add (-a31/a11)*R1 to row 3 to get the zero value. Again, in this process, we consider a11 as the pivot element.

Practice-problem-2-find x, y and z value for practice problem 38.

Use a22 as a pivot to eliminate a32.

First, divide R2 by 5 to let the pivot a22 equal 1; also, divide R3 by 5 to minimize the numbers.

Multiply the second row by -7 and add the result to the third row. divide the third row by -88 to make a33 equals 1.

make a22 equals one by diving the second row by 5, and use as a pivot.

Perform the row operations to obtain the reduced echelon form, or simply an upper triangular matrix with three ones on the diagonal.

Use a33 as a pivot to eliminate a23& a13.

Similarly, we use the element a33 to eliminate a13 by adding (-a13/a33)*R3 to R1 to obtain the zero value. Again, in this process, we consider a33 as the pivot element. We have now obtained the required diagonal matrix with 1s on the diagonal; please refer to the next slide for clarity.

Use a33 as a pivot and eliminate a13 and a23.

Use a22 as a pivot to eliminate a12.

We will use the element a22 to eliminate a12; this is done by adding(-a12/a22)*R2 to R1. We have now obtained the required reduced row echelon form (Gauss-Jordan). Please refer to the next slide for clarity.

calculation for x -y and z for the practice problem-38.

We can then obtain the values Z = -2, Y = 1, and X = +1. In this method, we do not need to make back substitutions. However, it is important to check the Values of X, Y, and Z in any of the three equation systems to ensure that the solution is accurate.