Well, I'll respond to the question hoping it will help someone else in the future. The reason why you're having difficulty setting up the matrix is because you're too focused on the actual colors rather than the three mixes.

We want a total of 244 gallons of paint with:

50% red = 122 gallons of the initial 244

25% green = 61 gallons of the initial 244

25% blue = 61 gallons of the initial 244.

Don't give the variable names to the colors (we know how much we want) but rather to mix 1, mix 2, and mix 3:

MIXES RED GREEN BLUE

x = mix 1 .125x .75x .125x

y = mix 2 .2y .2y .6y

z = mix 3 (2/3)z (1/6)z (1/6)z

TOTAL 122 61 61

Thus your matrix should be:

Red: .125x + .2y + (2/3)z = 122

Green: .75x + .2y + (1/6)z = 61

Blue: .125x + .6y + (1/6) = 61

.125 .2 2/3 | 122

.75 .2 1/6 | 61

.125 .6 1/6 | 61

Solving for this gives: 32 gallons of mix 1; 50 gallons of mix 2, and 162 gallons of mix 3.