4.3 Gauss–Jordan Elimination

Question 1

Just as for systems of two linear equations in two variables, any linear system, regardless of the number of equations or number of variables, has either.

Answer 1

1. Exactly one solution (consistent and independent), or
2. Infinitely many solutions (consistent and dependent), or
3. No solution (inconsistent).

Question 2

What is a reduced matrix?

Answer 2

For large linear systems, it is not practical to list all such simplified forms; there are
too many of them. Instead, we give a general definition of a simplified form called a reduced matrix, which can be applied to all matrices and systems, regardless of size.

Question 3

Definition of Reduced form

Answer 3

Question 4

Examples of matrices in reduced form.

Answer 4

Question 5

What is a reduced system?

Answer 5

The system corresponding to a reduced augmented matrix is called a reduced system. As we shall see, reduced systems are easy to solve.

Question 6

PROCEDURE Gauss–Jordan Elimination

Answer 6

Step 1 Choose the leftmost nonzero column and use appropriate row operations to get a 1 at the top.

Step 2 Use multiples of the row containing the 1 from step 1 to get zeros in all remaining places in the column containing this 1

Step 3 Repeat step 1 with the submatrix formed by (mentally) deleting the row
used in step 2 and all rows above this row.

Step 4 Repeat step 2 with the entire matrix, including the rows deleted mentally.

Continue this process until the entire matrix is in reduced form.

Question 7

When is a system “Dependent”? What does that mean?

Answer 7

If the number of leftmost 1’s in a reduced augmented coefcient matrix is
less than the number of variables in the system and there are no contradic-tions, then the system is dependent and has infnitely many solutions.