Short Answer 
The problem involves defining the dimensions of a plot of land and formulating two linear equations based on given conditions. The equations are expressed in matrix form, and by calculating the inverse of the coefficient matrix, the values of length (x) and breadth (y) are determined.
Step 1: Formulate Linear Equations
We start by defining the dimensions of the plot of land. Let the length be x meters and the breadth be y meters. Based on the problem’s conditions, we can formulate two linear equations:
- From the first condition: If the length decreases by 20 meters and the breadth increases by 10 meters, the equation simplifies to x – y = 50.
- From the second condition: If the length decreases by 20 meters and the breadth increases by 10 meters, it simplifies to x – 2y = 0.
Step 2: Express the System in Matrix Form
Next, we can express the system of equations derived from the previous step in matrix form. This involves the matrix of coefficients, variables, and constants:
- Coefficients matrix A: [1 -1; 1 -2]
- Variables matrix X: [x; y]
- Constants matrix B: [50; 0]
This leads to the equation AX = B which we will solve next.
Step 3: Solve for Variables Using the Matrix Method
To find the values of x and y, we need to calculate the inverse of matrix A and then multiply it by matrix B. First, we compute the determinant of A, which must be non-zero for the inverse to exist:
- Determinant: det(A) = -1, which is valid.
- Inverse of A: A-1 = [2 -1; -1 -1].
- Finally, multiplying A-1 by B gives X = [100; -50].
Thus, we obtain the values for x and y from this calculation.