Question

A school wants to allocate students into three clubs: Sports, Music and Drama, under following conditions:

• The number of students in Sports club should be equal to the sum of the number of students in Music and Drama club. 
• The number of students in Music club should be 20 more than half the number of students in Sports club. 
• The total number of students to be allocated in all three clubs are 180.

Find the number of students allocated to different clubs, using matrix method.

Answer 1

Let number of students in sports = x

Number of students in music = y

Number of students in drama = z

According to the question,

x = y + z

x − y − z = 0   ...(i)

Again,

y − 20 = `1/2x`

x − 2y + 40 = 0    ...(ii)

x + y + z = 180    ...(iii)

Then,

A = `[(1, -1, -1), (1, -2, 0), (1, 1, 1)]`

X = `[(x), (y), (z)]`

B = `[(0), (-40), (180)]`

By matrix method,

AX = B

⇒ X = A⁻¹B

or, `X = 1/(|A|) ("adj" A)B   ...["From"  A^(-1) = "adj A"/|A|]`

Here, |A| = 1(–2) + 1(1) – (1 +2)

= –2 + 1 – 3

= –4

adj A = `[(c_11, c_12, c_13), (c_21, c_22, c_23), (c_31, c_32, c_33)]^T`

= `[(-2, -1, 3), (0, 2, -2), (-2, -1, -1)]^T`

adj A= `[(-2, 0, -2), (-1, 2, -1), (3, -2, -1)]^T`

X = `1/-4[(-2, 0, -2), (-1, 2, -1), (3, -2, -1)][(0), (-40), (180)]`

`[(x), (y), (z)] = (-1)/4[(0 + 0 -360), (0 - 80 - 180), (0 + 80 - 180)]`

`[(x), (y), (z)] = (-1)/4[(-360), (-260), (-100)]`

`[(x), (y), (z)] = [(90), (65), (25)]`

X = 90, Y = 65, Z = 25

Number of students in the sports club = 90

Number of students in music club = 65

Number of students in drama club = 25