Determinant Value And Unique Solution To System Of Linear Eq

Determinant value and unique solution to system of linear equations

Use determinants to determine if a unique solution exists for each of the following systems of linear equations:

  • (a) 12x₁ + 7x₂ = 147 and 15x₁ + 19x₂ = 168
  • (b) 2x₁ + 3x₂ = 27 and 6x₁ + 9x₂ = 81
  • (c) 4x₁ + 3x₂ + 5x₃ = 27, x₁ + 6x₂ + 2x₃ = 19, 3x₁ + x₂ + 3x₃ = 15
  • (d) 4x₁ + 2x₂ + 6x₃ = 28, 3x₁ + x₂ + 2x₃ = 4, x₁ + 5x₂ + 15x₃ = 70

Matrix Inversion in Market Equilibrium

Find the equilibrium prices for the following related markets involving two or three goods using matrix inversion techniques:

  • (a) 18P₁ − P₂ = 87 and −2P₁ + 36P₂ = 98
  • (b) 5P₁ − 2P₂ = 15 and −P₁ + 8P₂ = 16
  • (c) 11P₁ − P₂ − P₃ = 31, −P₁ + 6P₂ − 2P₃ = 26, and −P₁ − 2P₂ + 7P₃ = 24

Use the inverse of the matrix A = [[11, -1, -1], [-1, 6, -2], [-1, -2, 7]] which is given as A⁻¹ = [[...]] (details are provided in the problem statement).

Matrix Inversion in IS-LM Model

(a) Given the IS equation 0.3Y + 100r − 252 = 0, and the LM equation 0.25Y − 200r − 176 = 0, find the equilibrium level of national income Y and the interest rate r using matrix inversion techniques.

(b) Given the consumption function Y = C + I₀, where C = C₀ + bY, find the equilibrium income and household consumption Y and C using matrix inversion.

Cramer’s Rule

Use Cramer's rule to solve for the unknowns in each of the following systems:

  • (a) 7P₁ + 2P₂ = 60 and P₁ + 8P₂ = 78
  • (b) 0.4Y + 150r = 209 and 0.1Y − 250r = 35
  • (c) 5q₁ − 2q₂ + 3q₃ = 16, 2q₁ + 3q₂ − 5q₃ = 2, and 4q₁ − 5q₂ + 6q₃ = 7

Paper For Above instruction

Linear algebra techniques, including determinants, matrix inversion, and Cramer's rule, play a fundamental role in solving various systems of equations that arise in economic analysis. These mathematical tools are especially crucial in determining the existence of unique solutions, solving for equilibrium states in markets, and analyzing macroeconomic models such as the IS-LM model.

Determinants and Existence of Unique Solutions

To verify whether a system of linear equations has a unique solution, the determinant of its coefficient matrix must be non-zero. For example, consider the systems in parts (a) through (d). In part (a), the coefficient matrix is:

 |12  7 |

|15 19|

The determinant is calculated as (12)(19) − (7)(15) = 228 − 105 = 123, which is non-zero, indicating a unique solution exists.

In part (b), the matrix is:

 |2  3 |

|6  9|

The determinant is (2)(9) − (3)(6) = 18 − 18 = 0, indicating this system either has infinitely many solutions or no solution, but not a unique one.

For systems with three variables such as in part (c), the determinant involves a 3x3 matrix. The calculation for such determinants follows standard rules, and if the determinant is zero, the system lacks a unique solution.

The key insight here is that the determinant serves as a criterion for the solution's uniqueness, which is vital for stability analysis and predicting precise outcomes in economic models.

Matrix Inversion and Market Equilibria

Market equilibrium prices for goods involve solving linear systems that can be represented in matrix form. For the two-goods example in part (a), the system is:

18P₁ − P₂ = 87

−2P₁ + 36P₂ = 98

Expressed in matrix notation, A * P = B, where

A = [[18, -1],

[-2, 36]]

and

B = [87, 98]

The solution involves finding P = A⁻¹ * B, where A⁻¹ is the inverse of matrix A. Utilizing the provided inverse, the prices P₁ and P₂ can be calculated straightforwardly. Similar procedures apply to parts (b) and (c), involving 3x3 systems.

In macroeconomic contexts such as the IS-LM model, matrix inversion helps determine the equilibrium income level and interest rates by representing the model's equations in matrix form. For example, the equations:

0.3Y + 100r − 252 = 0
0.25Y − 200r − 176 = 0

can be written as matrix equations and inverted to find Y and r, providing insights into policy impacts and macroeconomic stability.

Application of Cramer's Rule in Economics

Cramer's rule provides an explicit solution for linear systems with non-zero determinants of the coefficient matrix. For the two-good price system, the coefficient matrix is:

 |7   2 |

|1   8|

The determinant is 78 − 21 = 56 − 2 = 54, which is non-zero, allowing the solution for P₁ and P₂ via determinants of matrices formed by replacing respective columns with the constants.

This approach extends to macroeconomic models and multivariate economic systems, enabling precise calculations of equilibrium variables like income, interest rates, or quantities of goods.

Conclusion

Mathematical methods such as determinants, matrix inversion, and Cramer's rule are indispensable in economic modeling. They offer rigorous tools for solving systems of equations, assessing solution existence, and deriving equilibrium values in markets and macroeconomic frameworks. Mastery of these methods enhances analytical clarity and underscores the quantitative nature of economic analysis.

References

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