Matrix – Definition & Types
A matrix is a rectangular array of numbers arranged in m rows and n columns → order m × n
A = [aij]m×n
A = [aij]m×n
Zero/Null Matrix
All elements = 0
Square Matrix
Number of rows = Number of columns = n
Diagonal Matrix
aij = 0 if i ≠ j
Identity/Unit Matrix (I)
aij = 1 if i=j, 0 otherwise
Algebra of Matrices
Addition/Subtraction
Only if same order
C = A ± B → cij = aij ± bij
Multiplication
Possible only if columns of A = rows of B
(AB)ij = Σ aikbkj
Transpose & Special Matrices
Transpose A' : (i,j)th element of A = (j,i)th element of A'
Symmetric Matrix
A = A' → aij = aji
Skew-Symmetric
A' = –A → aij = –aji
∴ Diagonal elements = 0
Orthogonal Matrix
AA' = A'A = I
Minor, Cofactor & Rank
Minor Mij
Determinant after deleting ith row & jth column
Cofactor Cij
Cij = (–1)i+j Mij
Rank ρ(A)
Highest order of non-zero minor
• Rank unchanged by elementary transformations
• ρ(A) = ρ(A')
• No skew-symmetric matrix has rank 1
Inverse & System of Equations
AX = B → X = A–1B (if |A| ≠ 0)
A–1 = (adj A) / |A|
A–1 = (adj A) / |A|
Unique solution when |A| ≠ 0
Determinants – Minor & Cofactor
Minor Mij
Det after removing ith row & jth column
Cofactor Cij
(–1)i+j × Mij
Properties of Determinants
Must Remember Properties
- Rows ↔ Columns → value unchanged
- Interchange two rows/columns → sign changes
- Two identical rows/columns → det = 0
- Multiply row/column by k → det × k
- Two proportional rows/columns → det = 0
Cramer's Rule
For 3×3 system:
x = D₁/D, y = D₂/D, z = D₃/D (D ≠ 0)
x = D₁/D, y = D₂/D, z = D₃/D (D ≠ 0)
| Case | Solution |
|---|---|
| D ≠ 0 | Unique solution |
| D = 0, D₁=D₂=D₃=0 | Infinite solutions |
| D = 0, any Dᵢ ≠ 0 | No solution |
JEE Main Tips & Weightage
Extremely important chapter — 3–5 questions every year!
High Weightage Topics
- Symmetric & Skew-symmetric matrices
- Properties of determinants (especially row operations)
- Cramer's rule & consistency of system
- Rank of matrix (especially 3×3)
- Adjoint and inverse method
- Orthogonal matrix
Quick Tricks
- Skew-symmetric of odd order → det = 0
- If two rows same → det = 0
- Triangle determinant → product of diagonal
- For system: |A| decides everything!