Step-by-Step Solution
Topic: Matrix Method & Applications
(I) FORMULATING EQUATIONS
Original Area = xy.
Condition 1: (x – 50)(y + 50) = xy
xy + 50x – 50y – 2500 = xy
50x – 50y = 2500 (Divide by 50) ⇒ x – y = 50 …(1)
Condition 2: (x – 10)(y – 20) = xy – 5300
xy – 20x – 10y + 200 = xy – 5300
-20x – 10y = -5500 (Divide by -10) ⇒ 2x + y = 550 …(2)
(II) MATRIX FORM AX = B
From equations (1) and (2):
| 1 | -1 |
| 2 | 1 |
| x |
| y |
=
| 50 |
| 550 |
(III)(A) INVERSE OF MATRIX A
1. Determinant |A|:
|A| = (1)(1) – (-1)(2) = 1 + 2 = 3
2. Adjoint of A:
Swap diagonal elements, change signs of off-diagonal elements.
Adj A =
| 1 | 1 |
| -2 | 1 |
3. A-1 = (1/|A|) × Adj A:
A-1 = (1/3) × [Matrix above]
(III)(B) FINDING AREA
First, solve for X (x, y) using X = A-1B.
| x |
| y |
= (1/3)
| 1 | 1 |
| -2 | 1 |
| 50 |
| 550 |
Multiply matrices:
- x = (1/3) [ 1(50) + 1(550) ] = (1/3)[600] = 200 m
- y = (1/3) [ -2(50) + 1(550) ] = (1/3)[-100 + 550] = (1/3)[450] = 150 m
Area of Plot:
Area = x × y = 200 × 150
Area = 30,000 m2