System of Linear Equations

System of Linear Equations

This lesson covers the following key concepts:

• System as AX = B (matrix method)
• Solution using A⁻¹: X = A⁻¹B
• Cramer's rule for solving equations
• Consistency: unique, infinite, no solution
• Homogeneous vs non-homogeneous systems
• Criteria based on |A| and adj(A)·B
• Applications to real-world problems

Important Formulas

• $$AX = B \Rightarrow X = A^{-1}B \text{ if } |A| \neq 0$$
• $$\text{Cramer's: } x_i = \frac{|A_i|}{|A|}$$
• $$|A| \neq 0: \text{unique solution}$$
• $$|A| = 0, (\text{adj A})B = 0: \text{infinite solutions}$$
• $$|A| = 0, (\text{adj A})B \neq 0: \text{no solution}$$
• $$\text{Homogeneous: non-trivial solution iff } |A| = 0$$