Module 8 Quick Reference
Introduction to Matrices - One-Page Cheat Sheet
Matrix Operations
Addition/Subtraction
A + B = [aij + bij]
A - B = [aij - bij]
A - B = [aij - bij]
Must have same dimensions
Scalar Multiplication
kA = [k · aij]
Multiply each element by k
Matrix Multiplication
If A is m×n and B is n×p, then AB is m×p
(AB)ij = Row i of A · Column j of B
(AB)ij = Row i of A · Column j of B
Important: AB is defined only when columns of A = rows of B
Matrix multiplication is NOT commutative: AB ≠ BA
Matrix multiplication is NOT commutative: AB ≠ BA
Determinants
2×2 Determinant
det([a b; c d]) = ad - bc
Example: det([3 5; 2 4]) = (3)(4) - (5)(2) = 2
3×3 Determinant
Use cofactor expansion
or row reduction
or row reduction
Expand along row/column with most zeros
Key Properties
- det(I) = 1
- det(AB) = det(A) · det(B)
- det(kA) = kn · det(A) for n×n matrix
- Swapping rows changes sign
- Upper triangular: det = product of diagonal
Inverse Matrices
A matrix is invertible if and only if det(A) ≠ 0
If invertible: AA-1 = A-1A = I
If invertible: AA-1 = A-1A = I
2×2 Inverse Formula
For A = [a b; c d], if det(A) ≠ 0:
A-1 = (1/det(A)) · [d -b; -c a]
A-1 = (1/det(A)) · [d -b; -c a]
Steps: 1) Find det(A), 2) Swap diagonal (a↔d), 3) Negate off-diagonal (-b, -c), 4) Multiply by 1/det(A)
Gauss-Jordan Method
[A | I] → [I | A-1]
Use row operations to transform left side to I; right side becomes A-1
Inverse Properties
- (A-1)-1 = A
- (AB)-1 = B-1A-1 (reverse order)
- det(A-1) = 1/det(A)
Solving Matrix Equations
Matrix Equation AX = B
If A is invertible: X = A-1B
Steps
- 1. Check det(A) ≠ 0
- 2. Find A-1
- 3. Calculate X = A-1B
- 4. Verify: AX = B
System of Equations
Write as AX = B where:
- A = coefficient matrix
- X = variable matrix
- B = constant matrix
Cramer's Rule
Use determinants to solve systems. Works when det(A) ≠ 0
Formula
For AX = B:
x = det(Ax)/det(A)
y = det(Ay)/det(A)
z = det(Az)/det(A)
x = det(Ax)/det(A)
y = det(Ay)/det(A)
z = det(Az)/det(A)
Where: Ax is matrix A with x-column replaced by constants B, etc.
When to Use
- Best for: Finding one variable in small systems
- Requires: Square matrix with det(A) ≠ 0
- Alternative: Inverse matrix method for all variables
Quick Formula Summary
| Operation | Formula | Requirement |
|---|---|---|
| Addition A+B | [aij + bij] | Same dimensions |
| Scalar kA | [k · aij] | None |
| Multiplication AB | Row·Column | cols(A) = rows(B) |
| Det 2×2 | ad - bc | Square matrix |
| Inverse 2×2 | (1/det)[d -b; -c a] | det(A) ≠ 0 |
| Solve AX=B | X = A-1B | A invertible |
| Cramer's Rule | x = det(Ax)/det(A) | det(A) ≠ 0 |
Common Mistakes to Avoid
- Order matters: AB ≠ BA (multiplication not commutative)
- Dimensions: Always check before multiplying
- Determinant signs: Watch alternating +/- in cofactor expansion
- Inverse formula: It's [d -b; -c a], not [d b; c a]
- Scalar determinants: det(kA) = kndet(A), not k·det(A)
- Check invertibility: Always verify det(A) ≠ 0 before finding inverse
Quick Tips
For Determinants
- Use 2×2 formula directly
- For 3×3: expand along row/column with zeros
- Upper triangular: multiply diagonal
For Inverses
- Check det(A) ≠ 0 first!
- 2×2: use formula
- Larger: Gauss-Jordan
- Verify: AA-1 = I
Solving Systems
- One variable → Cramer's
- All variables → Inverse
- Always verify solution
Matrix Multiplication
- Check: cols(A) = rows(B)
- Result: rows(A) × cols(B)
- Row dot Column
- Order matters!
Special Matrices
| Type | Description | Example |
|---|---|---|
| Zero Matrix | All zeros | [0 0; 0 0] |
| Identity I | 1's on diagonal, 0's elsewhere | [1 0; 0 1] |
| Diagonal | Non-zero only on diagonal | [2 0; 0 3] |
| Upper Triangular | Zeros below diagonal | [1 2; 0 3] |
Test Day Checklist
Bring: Calculator, pencils, eraser
Know: 2×2 det formula, 2×2 inverse formula, matrix multiplication rule
Check: Dimensions, det ≠ 0, arithmetic, verify answers
Know: 2×2 det formula, 2×2 inverse formula, matrix multiplication rule
Check: Dimensions, det ≠ 0, arithmetic, verify answers
Problem-Solving Order
- Read problem carefully
- Identify what's asked (determinant, inverse, solve system?)
- Check requirements (dimensions compatible? det ≠ 0?)
- Choose best method
- Show all work
- Verify answer if time permits