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Matrix A:

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📐 Matrix Formulas & Rules

Matrix Addition / Subtraction

Matrices must have the same dimensions. Add/subtract element by element.

[A+B]ᵢⱼ = Aᵢⱼ + Bᵢⱼ

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

Matrix Multiplication

A must be m×n, B must be n×p. Result is m×p. Row × Column dot product.

[AB]ᵢⱼ = Σ Aᵢₖ × Bₖⱼ

Note: AB ≠ BA in general (not commutative!)

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

Transpose

[Aᵀ]ᵢⱼ = Aⱼᵢ (flip rows and columns)

[1 2 3]ᵀ = [1 4]
[4 5 6] [2 5]
[3 6]

Properties: (AB)ᵀ = BᵀAᵀ, (Aᵀ)ᵀ = A

Determinant

2×2: det(A) = ad − bc
[a b]
[c d]

3×3: Cofactor expansion along row 1:
det(A) = a₁₁·M₁₁ − a₁₂·M₁₂ + a₁₃·M₁₃
where Mᵢⱼ = minor (det of submatrix)

Inverse Matrix

A⁻¹ exists only if det(A) ≠ 0

2×2: A⁻¹ = (1/det) × [ d -b]
[-c a]

nxn: A⁻¹ = adj(A) / det(A)

Verify: A × A⁻¹ = I (identity matrix)

Trace & Rank

Trace = sum of diagonal elements = Σ Aᵢᵢ
tr([1 2]) = 1 + 4 = 5
([3 4])

Rank = number of linearly independent rows/columns
Rank ≤ min(rows, cols)

Special Matrices

Identity (I): diagonal = 1, rest = 0
Zero: all elements = 0
Symmetric: A = Aᵀ
Diagonal: non-zero only on main diagonal
Singular: det = 0 (no inverse exists)
Orthogonal: AᵀA = I (Aᵀ = A⁻¹)

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