Rre25. find an equation involving g, h, and k that makes this augmented matrix correspond to a consistent system: 2 4 1 4 7 g 0 3 5 h 2 5 9 k 3

Multiply the first row by - g/2 and add to the second row. The new second row is

... [0, - 2g, 3-g/2, 5-2g, h-7g/2]

Multiply the first row by - 1 and add to the third row. The new third row is

... [0, 1, 8, k-4, - 4]

Now, multiply the new second row by 1 / (2g) and add to the third row. The new third row is ...

... [0, 0, 8 + (3-g/2) / (2g), k-4 + (5-2g) / (2g), - 4 + (h-7g/2) / (2g) ]

This third row simplifies to

... [0, 0, (6+31g) / 4g, (5-10g+2gk) / (2g), (2h-23g) / (4g) ]

Normalizing each row so its pivot element is 1 and finishing the row operations necessary to leave the left three columns a diagonal matrix puts terms in the right two columns that have 6+31g in their denominator.

The equations necessary to make the system consistent appear to be

... g ≠ 0

... 6 + 31g ≠ 0