Matrix Addition (same dimensions):
  [A + B]ᵢⱼ = Aᵢⱼ + Bᵢⱼ

Example:
  [1 2]   [5 6]   [6  8]
  [3 4] + [7 8] = [10 12]

Scalar Multiplication:
  [cA]ᵢⱼ = c × Aᵢⱼ

  3 × [1 2] = [3  6]
      [3 4]   [9 12]

Properties:
  A + B = B + A  (commutative)
  c(A + B) = cA + cB
  A + 0 = A  (zero matrix identity)

Addition is element-wise and commutative. Matrices must have identical dimensions. The zero matrix acts as the additive identity.

C = AB where Cᵢⱼ = Σₖ Aᵢₖ × Bₖⱼ

Dimensions: (m×p)(p×n) → (m×n)
Inner dimensions must match!

[1 2] × [5 6] = [1×5+2×7  1×6+2×8]
[3 4]   [7 8]   [3×5+4×7  3×6+4×8]
                = [19 22]
                  [43 50]

Complexity: O(n³) for n×n matrices
Strassen: O(n^2.807)
Best known: O(n^2.3728...)

AB ≠ BA  (NOT commutative!)
A(BC) = (AB)C  (associative)
AI = IA = A  (identity)

Matrix multiplication is row-by-column dot products. Non-commutativity is fundamental — rotation order in 3D graphics depends on this.

2×2:
  det[a b] = ad − bc
     [c d]

  det[3 7] = 3×2 − 7×1 = −1
     [1 2]

3×3 (cofactor expansion):
  det[a b c]
     [d e f] = a(ei−fh)−b(di−fg)+c(dh−eg)
     [g h i]

  det[1 2 3]
     [4 5 6] = 1(5·9−6·8)−2(4·9−6·7)
     [7 8 9]   +3(4·8−5·7)
             = 1(−3)−2(−6)+3(−3)
             = −3+12−9 = 0

det=0 → singular (no inverse)
det(AB) = det(A) × det(B)

The determinant tells you if the matrix is invertible (det≠0) and the volume scaling factor of the linear transformation. A zero determinant means the transformation 'squishes' space.

For 2×2:
  A = [a b]  →  A⁻¹ = (1/det) × [ d -b]
      [c d]                       [-c  a]

Example:
  A = [4 7]  det = 4×2−7×3 = −13
      [3 2]

  A⁻¹ = (1/−13) [ 2 -7] = [-2/13  7/13]
                [-3  4]   [ 3/13 -4/13]

Verify: A × A⁻¹ = I
  [4 7] [-2/13  7/13]
  [3 2] [ 3/13 -4/13]
  = [1 0]
    [0 1] ✓

For n×n: use Gauss-Jordan
elimination or adjugate method.

A matrix has an inverse if and only if det(A) ≠ 0. The inverse is crucial for solving systems Ax = b → x = A⁻¹b.