Solving Problems with Matrices
CSEC Mathematics: Real-World Matrix Applications
Essential Understanding: Matrices aren't just abstract mathematical objects—they're powerful tools for organizing and manipulating data. In CSEC, you'll encounter matrices in data representation, matrix multiplication, equality problems, and real-world applications like inventory management and cost calculations.
Matrix Multiplication
Matrix multiplication is different from scalar multiplication. To multiply two matrices, we use a "row by column" method.
Rule for Matrix Multiplication
Two matrices can be multiplied only if the number of columns in the first matrix equals the number of rows in the second matrix.
The red numbers must match!
How to Multiply Matrices
Row × Column Method
Each element in the result is found by multiplying a row from the first matrix by a column from the second matrix, then adding the products.
Worked Example 1: 2×2 Matrix Multiplication
Calculate: \(\begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \times \begin{pmatrix} 5 & 2 \\ 3 & 1 \end{pmatrix}\)
Result will be 2×2.
\((2 \times 5) + (3 \times 3) = 10 + 9 = 19\)
\((2 \times 2) + (3 \times 1) = 4 + 3 = 7\)
\((1 \times 5) + (4 \times 3) = 5 + 12 = 17\)
\((1 \times 2) + (4 \times 1) = 2 + 4 = 6\)
Answer: \(\begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \times \begin{pmatrix} 5 & 2 \\ 3 & 1 \end{pmatrix} = \begin{pmatrix} 19 & 7 \\ 17 & 6 \end{pmatrix}\)
⚠️ Important: Matrix Multiplication is NOT Commutative!
\(AB \neq BA\) in general
The order matters! Multiplying A × B gives a different result than B × A (if B × A is even possible).
Worked Example 2: Row Matrix × Column Matrix
Calculate: \(\begin{pmatrix} 3 & 2 & 4 \end{pmatrix} \times \begin{pmatrix} 1 \\ 5 \\ 2 \end{pmatrix}\)
Result will be 1×1 (a single number!)
Answer: \(\begin{pmatrix} 21 \end{pmatrix}\) or simply 21
Matrix Equality and Finding Unknowns
Two matrices are equal if they have the same order AND all corresponding elements are equal.
Matrix Equality
Worked Example 3: Finding Unknown Values
Given: \(\begin{pmatrix} 2x & 3 \\ 4 & y+1 \end{pmatrix} = \begin{pmatrix} 10 & 3 \\ 4 & 7 \end{pmatrix}\), find x and y.
Position (1,1): \(2x = 10\)
Position (2,2): \(y + 1 = 7\)
Answer: x = 5, y = 6
Data Matrices: Real-World Applications
Matrices are excellent for organizing and manipulating data. CSEC frequently tests your ability to interpret and work with data matrices.
📊 Application: Inventory Management
A shop has inventory at two branches represented by this matrix:
| Shirts | Pants | Jackets | |
|---|---|---|---|
| Branch A | 50 | 30 | 20 |
| Branch B | 40 | 45 | 15 |
This can be written as matrix: \(I = \begin{pmatrix} 50 & 30 & 20 \\ 40 & 45 & 15 \end{pmatrix}\)
Worked Example 4: Sales Analysis
A company has sales data for two products over three months:
Sales Matrix S: \(\begin{pmatrix} 100 & 150 & 200 \\ 80 & 120 & 160 \end{pmatrix}\) (Rows: Products A, B; Columns: Jan, Feb, Mar)
Prices: Product A costs $5, Product B costs $8
Find the total revenue for each month.
February: \(5(150) + 8(120) = 750 + 960 = 1710\)
March: \(5(200) + 8(160) = 1000 + 1280 = 2280\)
Answer: Revenue = \(\begin{pmatrix} 1140 & 1710 & 2280 \end{pmatrix}\)
January: $1,140, February: $1,710, March: $2,280
Worked Example 5: Cost Calculation (CSEC-Style)
A bakery makes cakes and cookies. The ingredient requirements (in kg) per batch are:
| Flour | Sugar | Butter | |
|---|---|---|---|
| Cakes | 2 | 1 | 0.5 |
| Cookies | 1 | 0.5 | 0.25 |
Flour costs $3/kg, Sugar costs $2/kg, Butter costs $5/kg.
(a) Write the ingredient matrix R and cost matrix C.
(b) Calculate the cost per batch of each product.
\(C = \begin{pmatrix} 3 \\ 2 \\ 5 \end{pmatrix}\) (3×1 - column matrix)
Cakes: \(2(3) + 1(2) + 0.5(5) = 6 + 2 + 2.5 = 10.50\)
Cookies: \(1(3) + 0.5(2) + 0.25(5) = 3 + 1 + 1.25 = 5.25\)
Answer: Cost per batch: Cakes = $10.50, Cookies = $5.25
Identity Matrix Properties
The identity matrix (I) is special because multiplying any matrix by I leaves it unchanged.
Identity Matrix Property
The 2×2 identity matrix: \(I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)
Example: Multiplying by Identity
\(\begin{pmatrix} 3 & 5 \\ 2 & 4 \end{pmatrix} \times \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 3 & 5 \\ 2 & 4 \end{pmatrix}\)
The result is the same as the original matrix!
CSEC Past Paper Style Questions
Test Your Understanding
Position (1,1): \(2(1) + 1(2) = 4\)
Position (1,2): \(2(0) + 1(3) = 3\)
Position (2,1): \(3(1) + 4(2) = 11\)
Position (2,2): \(3(0) + 4(3) = 12\)
From position (1,1): x = 3
From position (2,2): y = 5
Therefore: x + y = 3 + 5 = 8
(2 × 3) × (3 × 4) = (2 × 4)
The inner dimensions (3) match, so multiplication is possible.
The result takes the outer dimensions: 2 × 4
Tuesday sales: 40 pens, 35 notebooks
Revenue = \(40 \times \$2 + 35 \times \$5 = \$80 + \$175 = \$255\)
First expand: \(\begin{pmatrix} 2a & 6 \\ 2 & 2b \end{pmatrix} + \begin{pmatrix} 1 & -2 \\ 4 & 3 \end{pmatrix} = \begin{pmatrix} 7 & 4 \\ 6 & 9 \end{pmatrix}\)
Then: \(\begin{pmatrix} 2a+1 & 4 \\ 6 & 2b+3 \end{pmatrix} = \begin{pmatrix} 7 & 4 \\ 6 & 9 \end{pmatrix}\)
From (1,1): \(2a + 1 = 7 \Rightarrow a = 3\)
From (2,2): \(2b + 3 = 9 \Rightarrow b = 3\)
CSEC Examination Tips
- Matrix multiplication: Always check if multiplication is possible by comparing inner dimensions first.
- Order of result: For (m×n) × (n×p), the result is always (m×p).
- Non-commutativity: Remember AB ≠ BA. This is a favorite exam concept!
- Data matrices: Pay attention to what rows and columns represent. This context is often tested.
- Combined operations: Do scalar multiplication first, then addition/subtraction, then matrix multiplication (if applicable).
- Finding unknowns: Set up equations by equating corresponding elements. Solve systematically.
Summary: Matrix Problem-Solving Checklist
- ✅ Check matrix compatibility for the required operation
- ✅ For multiplication: inner dimensions must match
- ✅ Work systematically through each element
- ✅ Double-check signs when adding/subtracting
- ✅ For word problems: identify what rows and columns represent
- ✅ Label your matrices clearly in your working