The row reduction of … cf shows that … must be nonzero, since f and g are arbitrary. Otherwise, for some choices of f and g the second row could correspond to an equation of

Problem:

The row reduction of

   

to

   

shows that

   

must be nonzero, since

   

and

   

are arbitrary. Otherwise, for some choices of

   

and

   

, the second row could correspond to an equation of the form

   

, where

   

is nonzero. Thus,

   

.

Solution:

1. Start with the original matrix:

      

2. Apply row operations to reduce the matrix:
   • Make the first pivot element (top left) equal to 1. This is already satisfied in the given matrix.
   • Eliminate the second-row element below the pivot (in the same column).
3. After performing the necessary row operation (subtracting a multiple of the first row from the second), the reduced form of the matrix becomes:

      

4. Analyze the condition for consistency:
   • The term

        

     must be nonzero to avoid the second row becoming inconsistent (e.g., leading to an equation like

        

     where

        

     is nonzero).
   • If

        

     , the system might lose its uniqueness or consistency depending on the values of

        

     and

        

     .
5. Conclusion: For the system to remain consistent and valid, it is required that:

      

   .

Reference: This problem is adapted from Linear Algebra with Applications by David Lay.