Matrix and Determinant

Matrix

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices are fundamental in mathematics, physics, computer science, and data analysis.

Notation: Denoted by capital letters (A, B, C).

Order/Dimension: A matrix with m rows and n columns is an m × n matrix.

Example: $ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \quad \text{(2×3 matrix)} $

Types of Matrices:

• Square Matrix: Same number of rows and columns (m = n).
• Row Matrix: 1 × n.
• Column Matrix: m × 1.
• Zero Matrix: All elements are zero.
• Identity Matrix (I): Square matrix with 1s on the main diagonal and 0s elsewhere.

Transpose of a Matrix

The transpose of a matrix A (denoted as Aᵀ) is obtained by interchanging its rows and columns.

If A is m × n, then Aᵀ is n × m.

Example: $ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}, \quad A^T = \begin{bmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \end{bmatrix} $

Properties of Transpose of Matrix

Let A and B be matrices of appropriate orders, and k be a scalar.

• (Aᵀ)ᵀ = A (Transpose is involutory)
• (A + B)ᵀ = Aᵀ + Bᵀ
• (kA)ᵀ = k(Aᵀ)
• (AB)ᵀ = BᵀAᵀ (Reverse order for product)
• (Aᵀ)⁻¹ = (A⁻¹)ᵀ (if A is invertible)

Multiplication of Matrices

Two matrices can be multiplied only if the number of columns in the first equals the number of rows in the second.

If A is m × p and B is p × n, then AB is m × n.

The element at (i,j) in AB is the dot product of the i-th row of A and j-th column of B.

Example: $ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} $

$ AB = \begin{bmatrix} 1\cdot5 + 2\cdot7 & 1\cdot6 + 2\cdot8 \\ 3\cdot5 + 4\cdot7 & 3\cdot6 + 4\cdot8 \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix} $Note: Matrix multiplication is not commutative (AB ≠ BA in general).

Properties Related to Matrix Multiplication

• Associative: (AB)C = A(BC)
• Distributive: A(B + C) = AB + AC and (B + C)A = BA + CA
• Identity: AI = IA = A (where I is the identity matrix)
• Zero Matrix: A·0 = 0·A = 0 (zero matrix)
• Transpose of Product: (AB)ᵀ = BᵀAᵀ
• Scalar Multiplication: k(AB) = (kA)B = A(kB)

Determinant of a Matrix

The determinant (det(A) or |A|) is a scalar value computed from a square matrix only. It provides important information about the matrix (invertibility, volume scaling, etc.).

For 2×2 Matrix: $ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \quad \det(A) = ad - bc$

Key Properties:

• det(AB) = det(A)·det(B)
• det(Aᵀ) = det(A)
• det(kA) = kⁿ det(A) (n = order of matrix)
• If det(A) = 0, matrix is singular (not invertible).
• If det(A) ≠ 0, matrix is non-singular (invertible).

Inverse Matrix

The inverse of a square matrix A (denoted A⁻¹) satisfies: $ A \cdot A^{-1} = A^{-1} \cdot A = I $

Existence: A must be square and det(A) ≠ 0.

Formula for 2×2 Inverse: $ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \quad A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} $

General Method (for larger matrices):

• Compute det(A)
• Find cofactor matrix
• Transpose the cofactor matrix (adjugate)
• A⁻¹ = (1/det(A)) × adj(A)

Properties of Inverse:

• (A⁻¹)⁻¹ = A
• (AB)⁻¹ = B⁻¹A⁻¹ (reverse order)
• (Aᵀ)⁻¹ = (A⁻¹)ᵀ
• (kA)⁻¹ = (1/k)A⁻¹