Matrices
1. What is a Matrix? - Array of Numbers
Matrix is rectangular arrangement of numbers in rows and columns inside brackets. Example 2×3 matrix has 2 rows 3 columns 6 elements. Order m×n means m rows n columns. Notation A=[a_ij] where i=row, j=column. Example A=[[1,2,3],[4,5,6]] order 2×3, a_12=2 (first row second column), a_21=4. Matrices store data like marks of students.
Matrix: rectangular array m×n, m rows n columns, order m×n, element a_ij i=row j=col. Example [[1,2,3],[4,5,6]] order 2×3, a_12=2. Used to store data.
• Matrix: Rectangular array of numbers in brackets
• Order m×n: m rows, n columns, total m×n elements
• Notation: A=[a_ij], i=row, j=column
• Example 2×3: [[1,2,3],[4,5,6]] a_12=2
• Use: Store data like student marks table
2. Types of Matrices - Zero, Row, Column, Square
Zero matrix all elements zero, can be any order. Row matrix only one row order 1×n like [5,2,8]. Column matrix only one column order m×1 like [[7],[-1],[4]]. Square matrix rows=columns n×n, diagonal elements important. Rectangular rows≠columns. Example 2×2 square [[3,1],[0,4]] diagonal 3 and 4.
Zero matrix all 0. Row 1×n, Column m×1, Square n×n rows=cols, Rectangular m≠n. Diagonal elements a_ii. Square matrix has main diagonal.
• Zero: All entries 0, any size
• Row: Only one row 1×n [5,2,8]
• Column: Only one column m×1 vertical
• Square: n×n equal rows cols, diagonal a_ii
• Rectangular: m≠n, rows≠cols
3. Diagonal, Scalar, Identity, Triangular
Diagonal matrix is square matrix where non-diagonal zero, only diagonal may be non-zero. Example diag[2,5,3]. Scalar matrix is diagonal where all diagonal equal like diag[4,4,4]=4I. Identity matrix I is scalar with 1s on diagonal, like I₂=[[1,0],[0,1]], I₃=[[1,0,0],[0,1,0],[0,0,1]]. Triangular: upper triangular non-zero on and above diagonal, lower below. These special matrices make calculations easy.
Diagonal: square non-diagonal 0. Scalar: diagonal all equal kI. Identity I diagonal 1. Upper triangular a_ij=0 for i>j, Lower triangular a_ij=0 for i
• Diagonal: Square, non-diagonal 0, e.g. diag[2,5,3]
• Scalar: Diagonal with all equal e.g. 4I
• Identity I: Scalar with 1s, I₂=[[1,0],[0,1]]
• Upper triangular: Below diagonal 0
• Lower triangular: Above diagonal 0
• I is multiplicative identity: AI=IA=A
4. Addition and Scalar Multiplication
Addition: Add corresponding elements, only same order matrices can add. Example [[1,2],[3,4]]+[[0,1],[2,0]]=[[1,3],[5,4]]. Properties: Commutative A+B=B+A, Associative (A+B)+C=A+(B+C), Identity A+0=A, Inverse A+(-A)=0. Scalar multiplication: Multiply every element by k, e.g. 2×[[1,2],[3,4]]=[[2,4],[6,8]]. Properties distributive k(A+B)=kA+kB.
Addition same order add entries. Commutative, Associative, A+0=A, A+(-A)=0. Scalar kA multiply each entry by k. Distributive k(A+B)=kA+kB, (k+m)A=kA+mA, k(mA)=(km)A, 1A=A.
• Addition: Same order, add corresponding entries
• Commutative: A+B=B+A, Associative: (A+B)+C=A+(B+C)
• Identity: A+0=A, Inverse: A+(-A)=0
• Scalar: kA = multiply each entry by k
• Properties: k(A+B)=kA+kB, (k+m)A=kA+mA, k(mA)=(km)A
• 1A=A, 0A=Zero matrix
5. Multiplication of Matrices - Row × Column
Multiplication not like addition. AB defined only if columns of A = rows of B. If A is m×n and B is n×p then AB is m×p. Element (i,j) of AB = dot product of i-th row of A and j-th column of B. Example A 2×3 B 3×2 product 2×2. Properties: Not commutative AB≠BA generally, Associative (AB)C=A(BC), Distributive A(B+C)=AB+AC, Identity AI=IA=A. Multiplication is main use for transformations.
AB defined if n_Acol = n_Brow. If A m×n B n×p AB m×p. (AB)_ij = Σ A_ik B_kj. Not commutative, Associative, Distributive, AI=A, IA=A. Order matters.
• Condition: Columns of A = Rows of B
• If A m×n B n×p → AB m×p
• (AB)_ij = Row i of A · Column j of B
• Example: 2×3 × 3×2 = 2×2
• Properties: AB≠BA generally, (AB)C=A(BC), A(B+C)=AB+AC
• AI=IA=A
6. Transpose of a Matrix - Flipping
Transpose A^T or A' is formed by interchanging rows and columns, first row becomes first column. Example A=[[1,2,3],[4,5,6]] A^T=[[1,4],[2,5],[3,6]]. Properties: (A^T)^T=A, (A+B)^T=A^T+B^T, (kA)^T=kA^T, (AB)^T=B^T A^T reverse order. Transpose useful for symmetric check.
Transpose A^T rows↔cols (a_ij)^T = a_ji. Properties: (A^T)^T=A, (A+B)^T=A^T+B^T, (kA)^T=kA^T, (AB)^T=B^T A^T. Order of transpose reverses multiplication.
• A^T: Rows become columns, (a_ij)^T=a_ji
• Example: [[1,2,3],[4,5,6]]^T=[[1,4],[2,5],[3,6]]
• Properties: (A^T)^T=A
• (A+B)^T=A^T+B^T, (kA)^T=kA^T
• (AB)^T=B^T A^T reverse order
• Use: Check symmetric
7. Symmetric and Skew-Symmetric
Symmetric: A^T=A, matrix equal to its transpose, so a_ij=a_ji, symmetric across diagonal. Example [[1,2,3],[2,4,5],[3,5,6]] symmetric. Skew-symmetric: A^T=-A, so a_ij=-a_ji and diagonal must be 0. Example [[0,2,-3],[-2,0,4],[3,-4,0]] skew. Every square matrix can be written as sum of symmetric and skew-symmetric: A=½(A+A^T)+½(A-A^T).
Symmetric: A^T=A a_ij=a_ji. Skew: A^T=-A a_ij=-a_ji diagonal 0. Any square A = (A+A^T)/2 symmetric + (A-A^T)/2 skew. Diagonal of skew is zero.
• Symmetric: A^T=A, a_ij=a_ji, mirror across diagonal
• Example [[1,2,3],[2,4,5],[3,5,6]] symmetric
• Skew: A^T=-A, a_ij=-a_ji, diagonal 0
• Example [[0,2,-3],[-2,0,4],[3,-4,0]] skew
• Any square: A = symmetric part + skew part
• Symmetric part (A+A^T)/2, Skew part (A-A^T)/2
8. Invertible Matrices and Real Life
Invertible: Square A has inverse B if AB=BA=I, B=A⁻¹. Only square matrices can be invertible, and determinant ≠0. Example [[1,2],[3,5]] inverse exists. If AB=I then BA=I. Inverse unique. Real life: Matrices used everywhere - computer graphics rotate images using matrix multiplication, Google search uses matrices, cryptography encodes messages, solving system of equations, stock market data, 3D games transform objects with matrices. Without matrices, no graphics.
Invertible: AB=BA=I B=A⁻¹, det≠0, unique, (AB)⁻¹=B⁻¹A⁻¹. Only square. Real: Computer graphics rotation, Google PageRank, cryptography, solving linear equations, engineering, AI data, 3D games, photo filters.
• Invertible: AB=BA=I, B=A⁻¹, det≠0, only square
• Unique inverse, (AB)⁻¹=B⁻¹A⁻¹
• Real graphics: Rotate, scale images via multiplication
• Google: PageRank uses matrices
• Crypto: Encode message via matrix
• Daily: Solve equations, 3D games, photo filters, data science
• Matrix: m×n array, a_ij row i col j
• Types: Zero all 0, Row 1×n, Column m×1, Square n×n, Rectangular m≠n, Diagonal non-diag 0, Scalar diag equal, Identity I diag 1, Upper/Lower triangular
• Addition: Same order add entries, Commutative A+B=B+A, Associative, A+0=A
• Scalar: kA multiply each entry, k(A+B)=kA+kB
• Multiplication: A m×n B n×p → AB m×p, (AB)_ij=Row·Col, AB≠BA, (AB)C=A(BC), (AB)^T=B^T A^T
• Transpose: Rows↔Cols, (A^T)^T=A, (A+B)^T=A^T+B^T, (AB)^T=B^T A^T
• Symmetric: A^T=A, Skew: A^T=-A diag 0, Any A=sym+skew
• Invertible: AB=BA=I, B=A⁻¹, det≠0, (AB)⁻¹=B⁻¹A⁻¹
• Real: Graphics rotation, Google, crypto, solving equations, 3D games, AI data