Determinants and Solving Linear Equation Systems (Economic Applications)

1) Determinant: meaning

A determinant is a scalar value computed from a square matrix; it helps determine:

• whether the matrix is invertible,
• whether a system of linear equations has a unique solution.

2) Determinant of 2×2 and 3×3

2.1 2×2

2.2 3×3 (concept: expansion)

For

One expansion:

3) Properties of determinants (high-yield)

1. If two rows (or columns) are identical → determinant = 0.
2. If a row (or column) is all zeros → determinant = 0.
3. Swapping two rows changes sign of determinant.
4. Multiplying a row by k multiplies determinant by k.
5. For triangular matrix, determinant = product of diagonal elements.
6. $|AB| = |A||B|$ (only when A and B are square of same order).