Determinants

Determinant is a scalar value derived from a square matrix. It is used to determine invertibility, solutions of equations, and area/volume calculations.

Key Concepts:

1. Determinant of 2×2 matrix: $$|A| = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad – bc$$∣A∣=​ac​bd​​=ad−bc
2. Determinant of 3×3 matrix (Sarrus Rule or cofactor expansion)
3. Properties:
   • If rows/columns are swapped → sign changes
   • If a row/column is multiplied by k → determinant multiplied by k
   • If two rows/columns are identical → determinant = 0
4. Applications: Solving linear equations (Cramer’s Rule), Eigenvalues, Area/Volume

Example Problem:$$A = \begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}, |A| = 1×4 – 2×3 = -2$$A=[13​24​],∣A∣=1×4−2×3=−2