Matrices

Definition of a Matrix

• A matrix is a rectangular arrangement of elements in rows and columns.
• Represented by capital letters A, B, C, …
• Elements are denoted by aᵢⱼ.

Order of a Matrix

• A matrix with m rows and n columns has order m × n.
• Total elements = mn.

Types of Matrices

• Row Matrix: Only one row (m = 1)
• Column Matrix: Only one column (n = 1)
• Rectangular Matrix: m ≠ n
• Square Matrix: m = n
• Singleton Matrix: Only one element (1 × 1)
• Zero (Null) Matrix: All elements are zero

Diagonal & Related Matrices

• Diagonal Matrix: aᵢⱼ = 0 for i ≠ j
• Scalar Matrix: Diagonal elements equal (aᵢᵢ = k)
• Unit Matrix (I): Diagonal elements = 1
• Number of zeros in diagonal matrix of order n: n² − n

Triangular Matrices

• Upper Triangular: aᵢⱼ = 0 for i > j
• Lower Triangular: aᵢⱼ = 0 for i < j
• Minimum number of zeros: n(n−1)/2

Trace of a Matrix

• Defined only for square matrices
• Trace(A) = a₁₁ + a₂₂ + … + aₙₙ
• Trace(A ± B) = Trace(A) ± Trace(B)

Equality of Matrices

• Two matrices are equal if:
• Same order and corresponding elements are equal

Addition & Subtraction of Matrices

• Defined only for matrices of same order
• (A ± B)ᵢⱼ = aᵢⱼ ± bᵢⱼ

Properties

• Commutative: A + B = B + A
• Associative: (A + B) + C = A + (B + C)
• Additive identity: A + O = A
• Additive inverse: A + (−A) = O

Scalar Multiplication

• kA = [kaᵢⱼ]
• k(A + B) = kA + kB
• (k + m)A = kA + mA
• k(mA) = (km)A

Multiplication of Matrices

• AB defined if columns of A = rows of B
• If A is m×n and B is n×p, then AB is m×p
• (AB)ᵢⱼ = Σ aᵢᵣbᵣⱼ

Properties

• Not commutative: AB ≠ BA
• Associative: (AB)C = A(BC)
• Distributive: A(B + C) = AB + AC
• Trace(AB) = Trace(BA)

Powers of a Matrix

• Defined only for square matrices
• AᵐAⁿ = Aᵐ⁺ⁿ
• (Aᵐ)ⁿ = Aᵐⁿ
• A⁰ = I

Transpose of a Matrix

• Interchanging rows and columns
• Order of Aᵀ is n×m if A is m×n
• (AB)ᵀ = BᵀAᵀ
• (Aᵀ)ᵀ = A

Symmetric & Skew-Symmetric Matrices

• Symmetric: Aᵀ = A
• Skew-symmetric: Aᵀ = −A
• Diagonal elements of skew-symmetric matrix are zero
• Any square matrix:
• A = ½(A + Aᵀ) + ½(A − Aᵀ)

Determinant of a Matrix

• Defined only for square matrices
• |AB| = |A||B|
• |Aᵀ| = |A|
• |kA| = kⁿ|A|
• Skew-symmetric matrix of odd order ⇒ determinant = 0

Adjoint of a Matrix

• Transpose of cofactor matrix
• A(adj A) = |A|I
• |adj A| = |A|ⁿ⁻¹
• adj(adj A) = |A|ⁿ⁻²A

Inverse of a Matrix

• Inverse exists iff |A| ≠ 0
• A⁻¹ = (1/|A|) adj A
• (AB)⁻¹ = B⁻¹A⁻¹
• (Aᵀ)⁻¹ = (A⁻¹)ᵀ

Special Matrices

• Orthogonal Matrix: AᵀA = I
• Idempotent Matrix: A² = A
• Involutory Matrix: A² = I
• Nilpotent Matrix: Aᵖ = 0

Rank of a Matrix

• Order of the highest non-singular square submatrix
• Rank of zero matrix = 0
• Rank(A) = Rank(Aᵀ)
• Elementary transformations do not change rank