Matrices and Determinants – JEE Mains Mathematics
1. Matrices
- A matrix is a rectangular array of numbers arranged in rows and columns.
- Order of a matrix: m × n (m rows and n columns)
2. Types of Matrices
- Square Matrix: Number of rows = number of columns
- Diagonal Matrix: All non-diagonal elements are zero
- Scalar Matrix: A diagonal matrix with equal diagonal elements
- Identity Matrix: A diagonal matrix with all diagonal elements = 1
- Zero Matrix: All elements are zero
- Row Matrix: Only one row
- Column Matrix: Only one column
- Symmetric and Skew-Symmetric Matrices
3. Algebra of Matrices
- Addition and Subtraction (only possible if matrices are of same order)
- Multiplication (A × B is defined only if the number of columns in A = number of rows in B)
- Scalar multiplication and Transpose of a Matrix
4. Determinants
- Defined only for square matrices
- Determinants of 2×2 and 3×3 matrices
- Used in finding area of triangle: If points are (x₁, y₁), (x₂, y₂), (x₃, y₃):
- Area = (1/2) × |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|
5. Properties of Determinants
- Interchanging rows/columns changes the sign
- Two identical rows/columns ⇒ determinant = 0
- If a row/column is multiplied by a scalar, determinant is also multiplied by the scalar
6. Adjoint and Inverse of a Matrix
- Adjoint: Transpose of the cofactor matrix
- Inverse of matrix A: A⁻¹ = adj(A)/|A| (only if |A| ≠ 0)
7. Solving System of Linear Equations
Using matrix method:
- AX = B ⇒ X = A⁻¹B, where A is coefficient matrix, X is variable matrix, B is constant matrix
- Applicable only if A⁻¹ exists (i.e., |A| ≠ 0)
- Used for 2 or 3 variable systems
8. Consistency of System
- Consistent: Has at least one solution
- Inconsistent: No solution
- Tested using rank or determinant