Introduction to Matrices

§ 01What is a Matrix?

A matrix is a rectangular arrangement of numbers, symbols, or expressions arranged in rows and columns. It is one of the central objects of linear algebra, compact, structured, and extraordinarily expressive.

We denote a matrix with an uppercase bold letter, such as A, and refer to its individual entries as a_ij where i is the row index and j is the column index (both starting from 1). A matrix with m rows and n columns is called an m × n matrix and is said to have size or dimension m × n.

A (2×3 matrix)

a₁₁a₁₂a₁₃ a₂₁a₂₂a₂₃

Row i, Column j → entry aij

Notice the convention: rows first, columns second. A 2×3 matrix has 2 rows and 3 columns. The entry a₂₃ is in the 2nd row and 3rd column. This row-then-column ordering is consistent throughout all matrix operations.

Why Matrices? Matrices compress an entire system of linear equations into a single object. The system 2x + 3y = 5, x − y = 1 becomes the matrix equation Ax = b where A = [[2,3],[1,−1]], x = [x,y]ᵀ, and b = [5,1]ᵀ. All the structure of the system lives in A.

§ 02Notation, Entries, and Dimension

The entry in row i and column j of matrix A is written a_ij or [A]_ij. We say two matrices are equal if and only if they have the same dimensions and every corresponding entry is equal.

Example 1Identifying entries in a 3×3 matrix

Let A = [[4,−1,2],[0,3,7],[−5,1,0]].

a₁₁ = 4 (row 1, col 1) · a₁₃ = 2 (row 1, col 3)
a₂₁ = 0 (row 2, col 1) · a₃₂ = 1 (row 3, col 2)
The main diagonal entries are a₁₁=4, a₂₂=3, a₃₃=0.

Dimension: 3 × 3 (square matrix)

Example 2Dimensions and equality

A 3×2 matrix and a 2×3 matrix are never equal, even if they contain the same numbers: their dimensions differ. Equality requires both same size AND matching entries.

A = B requires: same dimensions AND a_ij = b_ij for all i, j

§ 03Special Types of Matrices

Certain matrix structures appear so frequently they have dedicated names. Recognising them instantly saves time and unlocks shortcuts.

Square

m = n

Same number of rows and columns. Only square matrices have determinants and can be inverted.

Row Vector

1 × n

A single row. Written as [a b c].

Column Vector

m × 1

A single column. The standard form for solution vectors x in Ax = b.

Zero Matrix

All zeros

Every entry is 0. Plays the role of zero in matrix addition (additive identity).

Identity Matrix I

1s on diagonal

Square matrix with 1s on the main diagonal and 0s elsewhere. AI = IA = A always.

Diagonal

Off-diag = 0

Only the main diagonal entries can be non-zero. Easy to raise to powers: Dⁿ = diag(d₁ⁿ,…,dₙⁿ).

Upper Triangular

0s below diag

All entries below the main diagonal are zero. Row echelon form is upper triangular.

Symmetric

A = Aᵀ

Equal to its own transpose. Every entry a_ij = a_ji.

Identity I₃

100 010 001

Upper Triangular

314 027 005

Symmetric

123 256 369

§ 04Matrix Addition and Scalar Multiplication

These two operations are straightforward: they work entry-by-entry. The key constraint for addition is that both matrices must have identical dimensions.

Matrix Addition

[A + B]_ij = a_ij + b_ij

Add corresponding entries. Matrices must be the same size.

Example 3Add two 2×3 matrices

123456

70−12−34

822 6210

Each entry added position by position

Scalar Multiplication

[kA]_ij = k · a_ij

Multiply every entry by the scalar k. Any size matrix allowed.

Example 4Scalar multiplication: 3 × A

3 ×

2−104

6−3 012

Every entry multiplied by 3

§ 05Matrix Multiplication

Matrix multiplication is the defining operation of linear algebra, and the one that surprises students most. It is not entry-by-entry. Instead, each entry of the product is a dot product of a row from the left matrix with a column from the right matrix.

Matrix Multiplication Rule

[AB]_ij = Σ_k a_ik · b_kj

Row i of A dotted with column j of B gives entry (i,j) of AB.
Requires: #columns of A = #rows of B. Result size: (rows of A) × (cols of B).

Step-by-Step: Computing AB

Check compatibility: If A is m×n and B is n×p, AB is defined and has size m×p. If the inner dimensions don't match, the product is undefined.
For each entry (i,j) in the result: take row i of A and column j of B.
Compute the dot product: multiply corresponding entries and sum.
Place the result in position (i,j) of AB.

Row 2 of A (teal) · Column 3 of B (gold) → entry (2,3) of AB = cr + du

Example 52×2 matrix multiplication

A = [[1,2],[3,4]], B = [[5,6],[7,8]]. Find AB.

[AB]₁₁ = row 1 of A · col 1 of B = (1)(5)+(2)(7) = 5+14 = 19
[AB]₁₂ = row 1 · col 2 = (1)(6)+(2)(8) = 6+16 = 22
[AB]₂₁ = row 2 · col 1 = (3)(5)+(4)(7) = 15+28 = 43
[AB]₂₂ = row 2 · col 2 = (3)(6)+(4)(8) = 18+32 = 50

AB = [[19, 22], [43, 50]]

Example 6Non-square multiplication: (2×3)(3×2) → (2×2)

A = [[1,0,2],[−1,3,1]], B = [[3,1],[0,2],[4,−1]]. Inner dimensions both 3 ✓. Result is 2×2.

[AB]₁₁ = (1)(3)+(0)(0)+(2)(4) = 3+0+8 = 11
[AB]₁₂ = (1)(1)+(0)(2)+(2)(−1) = 1+0−2 = −1
[AB]₂₁ = (−1)(3)+(3)(0)+(1)(4) = −3+0+4 = 1
[AB]₂₂ = (−1)(1)+(3)(2)+(1)(−1) = −1+6−1 = 4

AB = [[11, −1], [1, 4]]

Matrix Multiplication is NOT Commutative In general, AB ≠ BA. Even when both products are defined, they usually give different results. Always check: "left times right" — the order matters. This is one of the most important differences between matrix arithmetic and ordinary number arithmetic.

Example 7AB ≠ BA: showing non-commutativity

A = [[1,2],[0,1]], B = [[1,0],[3,1]].

AB = [[1+6, 0+2],[0+3, 0+1]] = [[7,2],[3,1]]
BA = [[1+0, 2+0],[3+0, 6+1]] = [[1,2],[3,7]]

AB = [[7,2],[3,1]] ≠ BA = [[1,2],[3,7]]

§ 06The Transpose

The transpose of a matrix A, written Aᵀ, is formed by flipping the matrix over its main diagonal: rows become columns and columns become rows. If A is m×n, then Aᵀ is n×m.

Transpose Definition

[Aᵀ]_ij = a_ji

The entry in row i, column j of Aᵀ equals the entry in row j, column i of A.

Example 8Transpose of a 2×3 matrix

A (2×3)

123 456

→ Aᵀ =

Aᵀ (3×2)

14 25 36

Rows become columns, columns become rows

Key Transpose Properties (Aᵀ)ᵀ = A · (A+B)ᵀ = Aᵀ+Bᵀ · (kA)ᵀ = kAᵀ · (AB)ᵀ = BᵀAᵀ (order reverses!) · A is symmetric if A = Aᵀ.

§ 07The Matrix Equation Ax = b

The most important application of matrix multiplication is representing a linear system as a single equation Ax = b, where A is the coefficient matrix, x is the column vector of unknowns, and b is the right-hand side vector.

Example 9Convert a system to matrix form Ax = b

System: 2x + 3y − z = 4, x − y + 2z = −1, 3x + y + z = 7.

23−1 1−12 311

xyz

4−17

Three equations compressed into one matrix equation

Example 10Verify a solution to Ax = b

A = [[2,1],[1,3]], b = [5,10]ᵀ. Check whether x = [1,3]ᵀ is a solution.

Ax = [[2,1],[1,3]]·[1,3]ᵀ = [(2)(1)+(1)(3), (1)(1)+(3)(3)]ᵀ = [5,10]ᵀ = b ✓

x = [1, 3]ᵀ is a valid solution

§ 08The Matrix Inverse

For a square matrix A, its inverse A⁻¹ (if it exists) satisfies AA⁻¹ = A⁻¹A = I. The inverse exists if and only if det(A) ≠ 0.

To find the inverse systematically, augment A with the identity matrix and row-reduce until A becomes I: the right side simultaneously becomes A⁻¹.

2×2 Inverse Shortcut

[[a,b],[c,d]]⁻¹ = (1/(ad−bc)) · [[d,−b],[−c,a]]

Swap the main diagonal, negate the off-diagonal, divide by the determinant.

Example 11Find A⁻¹ for A = [[2, 5], [1, 3]]

det(A) = (2)(3)−(5)(1) = 6−5 = 1.
Swap diagonal, negate off-diagonal: [[3,−5],[−1,2]].
Divide by det = 1: A⁻¹ = [[3,−5],[−1,2]].
Verify: AA⁻¹ = [[2·3+5·(−1), 2·(−5)+5·2],[1·3+3·(−1), 1·(−5)+3·2]] = [[1,0],[0,1]] ✓

A⁻¹ = [[3, −5], [−1, 2]]

Example 12Row reduction to find inverse of 3×3

A = [[1,0,1],[0,2,1],[1,1,2]]. Augment: [A|I] and row reduce.

R3 ← R3 − R1: [A|I] bottom row becomes [0,1,1|−1,0,1]
R3 ← R3 − ½R2: bottom row → [0,0,½|−1,−½,1]
R3 ← 2R3: [0,0,1|−2,−1,2]
R2 ← R2−R3: [0,2,0|2,3,−2] → R2 ← ½R2: [0,1,0|1,3/2,−1]
R1 ← R1−R3: [1,0,0|3,1,−1]

A⁻¹ = [[3,1,−1],[1,1,−1],[−2,−1,2]] (verify: AA⁻¹ = I)

§ 09Laws of Matrix Algebra

Property	Formula	Note
Associativity of addition	(A+B)+C = A+(B+C)	Always holds
Commutativity of addition	A+B = B+A	Always holds
Associativity of multiplication	(AB)C = A(BC)	Always holds
Distributivity	A(B+C) = AB+AC	Always holds
Non-commutativity	AB ≠ BA in general	Critical difference from scalars
Transpose of product	(AB)ᵀ = BᵀAᵀ	Order reverses
Inverse of product	(AB)⁻¹ = B⁻¹A⁻¹	Order reverses
Zero product anomaly	AB = 0 does NOT require A=0 or B=0	Unlike scalar multiplication

§ 10Practice Quiz — Introduction to Matrices

10 questions covering dimensions, entry notation, addition, multiplication, transpose, and special types. Immediate feedback.

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