Step-by-Step Solution

Topic: Matrix Method for Linear Equations

(I) FORMULATING EQUATIONS

Total Donation = xy (constant)

Case 1: (x – 8)(y + 10) = xy

xy + 10x – 8y – 80 = xy   ⇒   10x – 8y = 80

Dividing by 2: 5x – 4y = 40 …(1)

Case 2: (x + 16)(y – 10) = xy

xy – 10x + 16y – 160 = xy   ⇒   -10x + 16y = 160

Dividing by 2: -5x + 8y = 80 …(2)

(II) MATRIX FORM AX = B

Writing the system in matrix form:

5 -4
-5 8
x
y

=

40
80

Where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

(III)(A) INVERSE OF MATRIX A

1. Determinant |A|:

|A| = (5)(8) – (-4)(-5) = 40 – 20 = 20

2. Adjoint of A:

Interchange diagonal elements, change signs of off-diagonal elements.

Adj A =

8 4
5 5

3. A-1 = (1/|A|) × Adj A:

A-1 = (1/20) × [Matrix above]

(III)(B) SOLVING FOR X AND Y

Using the formula X = A-1B:

x
y

  = (1/20)

8 4
5 5
40
80

Multiply matrices:

x = (1/20) [ 8(40) + 4(80) ] = (1/20)[320 + 320] = 640/20 = 32

y = (1/20) [ 5(40) + 5(80) ] = (1/20)[200 + 400] = 600/20 = 30

Number of Children (x) = 32
Donation per Child (y) = ₹ 30

⚠️ Silly Mistake Audit: Adjoint Calculation

When calculating Adj A for a 2×2 matrix, remember: Swap the diagonal elements (5 and 8) and Change the Sign of the off-diagonal elements (-4 becomes 4, -5 becomes 5). Do not swap the off-diagonal elements!

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