Matrices and Determinants – JEE Mains Mathematics

• by Zenith LMS
• Updated April 2025

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