Matrix Fundamentals
Matrix Definition
A matrix is a rectangular array of numbers arranged in rows and columns.
- An m × n matrix has m rows and n columns
- Square matrix: m = n (same number of rows and columns)
- Elements: denoted as aᵢⱼ where i = row, j = column
Special Matrices
- Identity Matrix (I): Square matrix with 1s on diagonal, 0s elsewhere
- Zero Matrix: All elements are 0
- Diagonal Matrix: Non-zero elements only on main diagonal
- Upper/Lower Triangular: Non-zero elements only above/below main diagonal
Determinants
For 2×2 Matrices
For matrix A = [a b; c d]: det(A) = ad - bc
For 3×3 Matrices
For matrix A = [a b c; d e f; g h i]: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) (remember the minus in the middle!)
Key Determinant Properties
- det(AB) = det(A) × det(B)
- det(Aᵀ) = det(A) (transpose doesn’t change determinant)
- det(kA) = kⁿdet(A) where n is matrix size and k is scalar
- If any row/column is all zeros → det(A) = 0
- Swapping two rows/columns → det changes sign
- Adding multiple of one row to another → det unchanged
Matrix Inverses
Invertible (Non-singular) Matrix Conditions
A matrix A is invertible if and only if:
- A is square (m × n where m = n)
- det(A) ≠ 0
- All rows/columns are linearly independent
- A⁻¹ exists such that AA⁻¹ = A⁻¹A = I
Singular (Non-invertible) Matrix
A matrix is singular if:
- Not square (different number of rows and columns)
- det(A) = 0 (even if square)
- Has linearly dependent rows/columns
Finding 2×2 Inverse
For A = [a b; c d], if det(A) ≠ 0: A⁻¹ = (1/det(A)) × [d -b; -c a]
Matrix Operations
Addition/Subtraction
- Only possible if matrices have same dimensions
- Add/subtract corresponding elements
Multiplication
- AB exists if columns of A = rows of B
- If A is m×n and B is n×p, then AB is m×p
- Not commutative: AB ≠ BA (usually)
- Associative: (AB)C = A(BC)
- Distributive: A(B + C) = AB + AC
Transpose
- Aᵀ: flip matrix over main diagonal
- (AB)ᵀ = BᵀAᵀ (reverse order!)
- (Aᵀ)ᵀ = A
Critical Rules for Problem Solving
Inverse Existence Checklist
- ✅ Is the matrix square? If no → not invertible
- ✅ Calculate determinant → if det = 0, not invertible
- ✅ If det ≠ 0 → matrix is invertible
Common Mistakes to Avoid
- Non-square matrices: Cannot be invertible (most common error)
- Determinant calculation errors: Double-check arithmetic
- Confusing singular vs non-singular:
- Singular = NOT invertible (det = 0)
- Non-singular = invertible (det ≠ 0)
Quick Determinant Checks
- Row/column of zeros → det = 0
- Two identical rows/columns → det = 0
- One row is multiple of another → det = 0
Practice Framework (Feynman Method)
Level 1: Can you explain to a child?
“A matrix is like a box of numbers. Some boxes can be ‘undone’ (have an inverse), others can’t.”
Level 2: Can you identify the patterns?
- Square + non-zero determinant = invertible
- Non-square OR zero determinant = not invertible
Level 3: Can you solve problems quickly?
- Check if square
- Calculate determinant
- Apply rules
Level 4: Can you teach it back?
Explain why these conditions matter and how they connect to linear independence and system solvability.
Quick Reference Formulas
| Concept | 2×2 Formula | Key Rule |
|---|---|---|
| Determinant | ad - bc | If det = 0, not invertible |
| Inverse | (1/det) × [d -b; -c a] | Only exists if det ≠ 0 |
| Multiplication | (AB)ᵢⱼ = Σ(AᵢₖBₖⱼ) | Columns of A = Rows of B |
Memory Aids
- “Square and Detective”: Must be square AND detective (determinant) ≠ 0
- “SODA”: Square → Only → Determinant → Acceptable (non-zero)
- “Non-square = Non-starter”: If not square, stop immediately - not invertible