Question

The management committee of a Welfare Club decided to award some of its members (say x) for sincerity, some (say y) for helping others selflessly and some others (say z) for effective management. The sum of all the awardees is 12. Three times the sum of all awardees for helping others selflessly and effective management added to two times the number of awardees for sincerity is 33. If the sum of the number of awardees for sincerity and effective management is twice the number of awardees for helping others, use matrix method to find the number of awardees of each category.

Answer 1

x + y + z = 12

2x + 3y + 3z = 33

x – 2y + z = 0

\(\begin{bmatrix}1&1&1\\2&3&3\\1&-2&1\end{bmatrix}\)\(\begin{bmatrix}x\\y\\z\end{bmatrix}\) = \(\begin{bmatrix}12\\33\\0\end{bmatrix}\)

|A| = 3 ≠ 0

adjA = \(\begin{bmatrix}9&-3&0\\1&0&-1\\-7&3&1\end{bmatrix}\)

X = A^-1B ⇒ \(\begin{bmatrix}x\\y\\z\end{bmatrix}\) = \(\frac{1}{3}\)\(\begin{bmatrix}9&-3&0\\1&0&-1\\-7&3&1\end{bmatrix}\)\(\begin{bmatrix}12\\33\\0\end{bmatrix}\)

X = A^-1B ⇒ \(\begin{bmatrix}x\\y\\z\end{bmatrix}\) = \(\frac{1}{3}\) \(\begin{bmatrix}108-99+0\\12+0-0\\-84+99+0\end{bmatrix}\)

= \(\frac{1}{3}\)\(\begin{bmatrix}9\\12\\15\end{bmatrix}\)

= \(\begin{bmatrix}3\\4\\5\end{bmatrix}\)

Hence x = 3 , y = 4, z = 5