The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others using matrix category. Apart from these values namely, honesty, cooperation and supervision, suggest one more value which the management of the colony must include for awards.

Given:
x + y + z = 12 (1)
3(y + z) + 2x = 33 (2)
(x + z) = 2y (3)
x + y + z = 12
2x + 3y + 3z = 33
x - 2y + z = 0
Form a matrix,
\begin{bmatrix} 1 & 1 & 1 \ 2 & 3 & 3 \ 1 & -2 & 1 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 12 \ 33 \ 0 \end{bmatrix}
AX = B
A⁻¹(AX) = A⁻¹(B)
IX = A⁻¹(B)
X = A⁻¹B
X = (Adj.A).B/|A|
|A| = \begin{vmatrix} 1 & 1 & 1 \ 2 & 3 & 3 \ 1 & -2 & 1 \end{vmatrix} = 1(3 + 6) - 1(2 - 3) + 1(-4 - 3) = 9 + 1 - 7 = 3
|A| ≠ 0
Adj.A = \begin{bmatrix} 9 & -3 & 0 \ 1 & 0 & -1 \ -7 & 3 & 1 \end{bmatrix}
(Adj.A).B = \begin{bmatrix} 9 & -3 & 0 \ 1 & 0 & -1 \ -7 & 3 & 1 \end{bmatrix}.\begin{bmatrix} 12 \ 33 \ 0 \end{bmatrix} = \begin{bmatrix} 9(12) -3(33) \ 12 \ -7(12) + 3(33) \end{bmatrix} = \begin{bmatrix} 108 - 99 \ 12 \ -84 + 99 \end{bmatrix} = \begin{bmatrix} 9 \ 12 \ 15 \end{bmatrix}
X = (Adj.A).B/|A|
X = (1/3)\begin{bmatrix} 9 \ 12 \ 15 \end{bmatrix} ⇒ X = \begin{bmatrix} 3 \ 4 \ 5 \end{bmatrix} ⇒ \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 3 \ 4 \ 5 \end{bmatrix}
So, x = 3, y = 4, z = 5
Therefore, the number of awardees for honesty, cooperation and supervision respectively are 3, 4 and 5.
Another value which management can include may be regularity and sincerity.