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Let $x$ be the number of students in the Sports club, $y$ be the number of students in the Music club, and $z$ be the number of students in the Drama club.
Based on the given conditions, we can formulate the following equations:\r\n\r\n1. The number of students in Sports club should be equal to the sum of the number of students in Music and Drama club: $x = y + z$\r\n2. The number of students in Music club should be 20 more than half the number of students in Sports club: $y = \frac{1}{2}x + 20$\r\n3. The total number of students to be allocated in all three clubs are 180: $x + y + z = 180$
Rewrite the equations in the standard form $ax + by + cz = d$:\r\n\r\n1. $x - y - z = 0$\r\n2. $-\frac{1}{2}x + y = 20$ or $-x + 2y = 40$\r\n3. $x + y + z = 180$
Express the system of equations as a matrix equation $AX = B$, where:\r\n\r\n$A = \begin{bmatrix} 1 & -1 & -1 \\ -1 & 2 & 0 \\ 1 & 1 & 1 \end{bmatrix}$, $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$, and $B = \begin{bmatrix} 0 \\ 40 \\ 180 \end{bmatrix}$
Calculate the determinant of matrix $A$:\r\n\r\n$|A| = \begin{vmatrix} 1 & -1 & -1 \\ -1 & 2 & 0 \\ 1 & 1 & 1 \end{vmatrix} = 1(2 - 0) - (-1)(-1 - 0) + (-1)(-1 - 2) = 2 - 1 + 3 = 4$\r\n\r\nSince $|A| \neq 0$, the system has a unique solution.
Find the adjoint of matrix $A$:\r\n\r\n$adj(A) = \begin{bmatrix} 2 & 0 & 2 \\ 1 & 2 & 1 \\ -3 & -2 & 1 \end{bmatrix}^T = \begin{bmatrix} 2 & 1 & -3 \\ 0 & 2 & -2 \\ 2 & 1 & 1 \end{bmatrix}$
Calculate the inverse of matrix $A$:\r\n\r\n$A^{-1} = \frac{1}{|A|} adj(A) = \frac{1}{4} \begin{bmatrix} 2 & 1 & -3 \\ 0 & 2 & -2 \\ 2 & 1 & 1 \end{bmatrix}$
Solve for $X$ using $X = A^{-1}B$:\r\n\r\n$X = \frac{1}{4} \begin{bmatrix} 2 & 1 & -3 \\ 0 & 2 & -2 \\ 2 & 1 & 1 \end{bmatrix} \begin{bmatrix} 0 \\ 40 \\ 180 \end{bmatrix} = \frac{1}{4} \begin{bmatrix} 2(0) + 1(40) - 3(180) \\ 0(0) + 2(40) - 2(180) \\ 2(0) + 1(40) + 1(180) \end{bmatrix} = \frac{1}{4} \begin{bmatrix} 40 - 540 \\ 80 - 360 \\ 40 + 180 \end{bmatrix} = \frac{1}{4} \begin{bmatrix} -500 \\ -280 \\ 220 \end{bmatrix} = \begin{bmatrix} -125 \\ -70 \\ 55 \end{bmatrix}$
There was an error in the calculation. Let's use the equations to solve for the variables using substitution or elimination.\r\n\r\nFrom equation (1): $x = y + z$\r\nFrom equation (3): $x + y + z = 180$\r\nSubstitute (1) into (3): $x + x = 180 \Rightarrow 2x = 180 \Rightarrow x = 90$\r\n\r\nFrom equation (2): $y = \frac{1}{2}x + 20 = \frac{1}{2}(90) + 20 = 45 + 20 = 65$\r\n\r\nFrom equation (1): $90 = 65 + z \Rightarrow z = 90 - 65 = 25$\r\n\r\nSo, $x = 90$, $y = 65$, and $z = 25$.
Final Answer: Sports: 90, Music: 65, Drama: 25
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