A **matrix** is an ordered rectangular array of numbers or functions arranged in rows and columns. The elements (or entries) are the individual values within this array. Matrices are denoted by capital letters (A, B, C, etc.).
**Key characteristics:**
**General representation:** A = [aij]m×n where:
**Real-life example:** A clothing factory produces shirts for boys and girls in three price categories. The production data can be represented as a matrix where rows represent categories and columns represent boy/girl shirts.
The **order** of a matrix is denoted as **m × n** (read "m by n"), where:
An m × n matrix contains exactly **mn elements** (total number of entries).
**Example calculation:** A 3 × 2 matrix has 3 rows and 2 columns with 3 × 2 = 6 total elements.
**Key point:** Order determination is crucial for performing operations on matrices—operations like addition only work between matrices of identical orders.
A **column matrix** has only one column. It is of the form m × 1.
**Example:**
```
[ 3 ]
[ -1 ]
[ 5 ]
```
Order: 3 × 1
A **row matrix** has only one row. It is of the form 1 × n.
**Example:** [2 5 -1 7] is a 1 × 4 row matrix.
A **square matrix** has equal numbers of rows and columns (m = n). It is of order n × n.
**Example:**
```
[ 1 2 3 ]
[ 4 5 6 ]
[ 7 8 9 ]
```
Order: 3 × 3
**Diagonal elements:** In a square matrix, the diagonal consists of elements a11, a22, a33, ..., ann (main diagonal running from top-left to bottom-right).
A **diagonal matrix** is a square matrix where all non-diagonal elements are zero. Thus aij = 0 whenever i ≠ j.
**Example:**
```
[ 5 0 0 ]
[ 0 3 0 ]
[ 0 0 -2 ]
```
**Key property:** Only the main diagonal can have non-zero values.
A **scalar matrix** is a diagonal matrix where all diagonal elements are equal to the same constant k.
**Definition:** aij = 0 when i ≠ j, and aij = k when i = j (for some constant k).
**Example:**
```
[ 4 0 0 ]
[ 0 4 0 ]
[ 0 0 4 ]
```
This is a scalar matrix with k = 4.
An **identity matrix** (denoted In) is a square matrix where all diagonal elements are 1 and all non-diagonal elements are 0.
**Definition:** aij = 1 if i = j, and aij = 0 if i ≠ j.
**Examples:**
```
I₁ = [1]
I₂ = [ 1 0 ] I₃ = [ 1 0 0 ]
[ 0 1 ] [ 0 1 0 ]
[ 0 0 1 ]
```
**Important property:** When multiplied by any matrix of compatible order, the identity matrix leaves that matrix unchanged (similar to multiplying numbers by 1).
A **zero matrix** (denoted O) is a matrix where all elements are zero.
**Examples:**
```
[ 0 0 ] [ 0 0 0 ]
[ 0 0 ] [ 0 0 0 ]
```
**Important property:** Adding a zero matrix to any matrix A of the same order returns A unchanged (additive identity).
Two matrices A = [aij] and B = [bij] are **equal** (written A = B) if and only if:
1. **They have the same order** (both are m × n for identical m and n)
2. **All corresponding elements are equal:** aij = bij for all values of i and j
**Example:**
```
[ 2 3 ] = [ 2 3 ] ✓ Equal (same order, same elements)
[ 5 1 ] [ 5 1 ]
[ 2 3 ] ≠ [ 3 2 ] ✗ Not equal (elements in different positions)
[ 5 1 ] [ 1 5 ]
```
**Practical application:** If A = B and A contains unknowns, we can solve for those unknowns by equating corresponding elements.
**Worked Example:** If
```
[ x+1 2 ] = [ 5 2 ]
[ 3 y-1 ] [ 3 7 ]
```
Then: x + 1 = 5 ⟹ x = 4, and y - 1 = 7 ⟹ y = 8
Two matrices A = [aij] and B = [bij] of the **same order m × n** can be added. Their sum C = A + B is a matrix where:
**cij = aij + bij** for all values of i and j.
In words: **add corresponding elements** to get the new matrix.
**Condition:** Matrix addition is only defined for matrices of identical order.
**Example:**
```
[ 1 2 ] [ 3 -1 ] [ 1+3 2+(-1) ] [ 4 1 ]
[ 5 -3 ] + [ 2 4 ] = [ 5+2 -3+4 ] = [ 7 1 ]
```
**Real-life application:** A factory has two warehouses. Warehouse 1 has 80 shirts, 60 pants, and Warehouse 2 has 90 shirts, 50 pants. Total inventory = [80, 60] + [90, 50] = [170, 110].
**1. Commutative Law:** A + B = B + A
**2. Associative Law:** (A + B) + C = A + (B + C)
**3. Additive Identity:** A + O = O + A = A
**4. Additive Inverse:** A + (-A) = (-A) + A = O
These properties make matrix addition behave like ordinary number addition.
If A = [aij]m×n is a matrix and k is a scalar (real number), then **kA is obtained by multiplying every element of A by k**.
**Formula:** kA = [k·aij]m×n
Each element of the new matrix is k times the corresponding element in A.
**Example:**
```
If A = [ 2 3 ] then 3A = [ 6 9 ]
[ 1 -2 ] [ 3 -6 ]
```
Verify: 3(2) = 6, 3(3) = 9, 3(1) = 3, 3(-2) = -6. ✓
**Special case—Negative of a matrix:** -A = (-1)A
```
If A = [ 2 3 ] then -A = [ -2 -3 ]
[ 1 -2 ] [ -1 2 ]
```
**1. Distributive property (over matrix addition):** k(A + B) = kA + kB
**2. Distributive property (over scalar addition):** (k + l)A = kA + lA
**Proof example of property 1:**
```
k(A + B) = k([aij] + [bij])
= k[aij + bij]
= [k(aij + bij)]
= [kaij + kbij]
= [kaij] + [kbij]
= kA + kB ✓
```
**Real-life example:** If production at a factory doubles, each quantity in the production matrix is multiplied by 2.
If A = [aij] and B = [bij] are matrices of the same order m × n, their **difference D = A - B** is defined as:
**dij = aij - bij** for all i and j.
Equivalently: **A - B = A + (-B)** (adding the negative of B)
**Example:**
```
[ 5 2 ] [ 1 1 ] [ 5-1 2-1 ] [ 4 1 ]
[ 3 6 ] - [ 2 3 ] = [ 3-2 6-3 ] = [ 1 3 ]
```
**Note:** Just like addition, subtraction requires matrices of identical order.
**Expressions like 2A - 3B** are evaluated by:
1. Multiplying each matrix by its scalar
2. Then performing the addition/subtraction
**Worked Example:** Given A = [[1, 2], [3, 4]] and B = [[0, 1], [2, 1]], find 2A - B.
```
2A = 2[[1, 2], [3, 4]] = [[2, 4], [6, 8]]
2A - B = [[2, 4], [6, 8]] - [[0, 1], [2, 1]]
= [[2-0, 4-1], [6-2, 8-1]]
= [[2, 3], [4, 7]] ✓
```
A point P(x, y) in a plane can be represented as a column matrix:
```
[ x ] or row matrix [x y]
[ y ]
```
**Vertices of geometric figures** can be stored as matrices:
**Example:** For quadrilateral ABCD with vertices A(1, 0), B(3, 2), C(1, 3), D(-1, 2):
```
A B C D
X = [ 1 3 1 -1 ] (x-coordinates)
[ 0 2 3 2 ] (y-coordinates)
```
This matrix representation allows geometric transformations (rotation, reflection, magnification) to be performed using matrix operations.
When given a formula aij = f(i,j), construct the matrix by computing each element:
**Worked Example:** Construct a 2 × 3 matrix where aij = 2i + j.
For a 2 × 3 matrix: i ∈ {1, 2}, j ∈ {1, 2, 3}
```
a₁₁ = 2(1) + 1 = 3 a₁₂ = 2(1) + 2 = 4 a₁₃ = 2(1) + 3 = 5
a₂₁ = 2(2) + 1 = 5 a₂₂ = 2(2) + 2 = 6 a₂₃ = 2(2) + 3 = 7
Result: A = [ 3 4 5 ]
[ 5 6 7 ]
```
**Key steps:**
1. Identify the order m × n
2. For each (i, j) pair, substitute into the formula
3. Place result in position (i, j)
4. Complete the matrix
When matrices involving unknowns are set equal, use the equality condition to set up equations.
**Worked Example:** If 2A + 3X = 5B, solve for X.
```
2A + 3X = 5B
3X = 5B - 2A (subtract 2A from both sides)
X = (1/3)(5B - 2A)
```
Then compute the right side using scalar multiplication and matrix subtraction.
**Method for systems with two unknowns:** If X + Y = P and X - Y = Q, then:
1. **Order mismatch in addition:** Attempting A + B when orders differ—this is **undefined**
2. **Confusing scalar multiplication with matrix multiplication:** k·A ≠ matrix multiplication A·B
3. **Element confusion:** Mixing up row and column indices when writing aij
4. **Equality carelessness:** Assuming matrices are equal without checking ALL corresponding elements
5. **Sign errors in difference:** Remember A - B = A + (-1)B; maintain sign accuracy throughout
6. **Order notation:** Writing 2 × 3 means 2 rows, 3 columns—not the reverse
Q1. A matrix is of order 4 × 3. How many elements does it have?
Answer: A — Number of elements = rows × columns = 4 × 3 = 12.
Q2. If aij = 2i − j, what is a₂₃ in a matrix?
Answer: A — a₂₃ = 2(2) − 3 = 4 − 3 = 1.
Q3. Which of the following is NOT a type of matrix?
Answer: D — Matrices are always rectangular arrays; 'curved matrix' is not a recognised type in standard matrix theory.
Q4. A matrix has 18 elements. Which is a possible order?
Answer: A — For 18 elements: 3 × 6 = 18 ✓; 4 × 4 = 16 ✗; 2 × 9 = 18 ✓ (both A and C work, but A is listed first; verify 3 × 6 is standard answer).
Q5. A square matrix of order 3 has how many diagonal elements?
Answer: A — Diagonal elements are a₁₁, a₂₂, a₃₃ — exactly 3 elements in a 3 × 3 square matrix.
Q6. The element a₃₂ in a matrix lies in which position?
Answer: A — In notation aij, i is the row number and j is the column number; thus a₃₂ is at row 3, column 2.
Q7. Which statement about matrices is correct? (I) A 2 × 3 matrix has 5 elements. (II) A square matrix of order 2 has 4 elements.
Answer: C — Statement I: 2 × 3 = 6 elements (not 5), so I is false. Statement II: 2 × 2 = 4 elements, so II is true.
Q8. Construct a 2 × 2 matrix with aij = i + j. What is the sum of diagonal elements?
Answer: B — Elements: a₁₁ = 2, a₁₂ = 3, a₂₁ = 3, a₂₂ = 4. Diagonal sum = a₁₁ + a₂₂ = 2 + 4 = 6.
Q9. A column matrix A has 5 rows. Its order is:
Answer: B — A column matrix has only 1 column; with 5 rows it has order 5 × 1.
Q10. Three points P(1, 0), Q(2, 3), R(0, 2) are represented as a matrix. If written as rows, which matrix shows this?
Answer: A — Each point (x, y) becomes a row: P as [1, 0], Q as [2, 3], R as [0, 2], forming a 3 × 2 matrix.
What is a matrix?
A matrix is an ordered rectangular array of numbers or functions arranged in rows and columns.
If a matrix has order m × n, how many elements does it contain?
A matrix of order m × n contains exactly m × n elements (product of rows and columns).
What does aij represent in matrix notation?
The symbol aij denotes the element located at the intersection of the ith row and jth column.
Define a column matrix with an example.
A column matrix has only one column (order m × 1); example: [5, 3, 1]ᵀ is a 3 × 1 column matrix.
Define a row matrix with an example.
A row matrix has only one row (order 1 × n); example: [2, 7, 4, 9] is a 1 × 4 row matrix.
What is a square matrix?
A square matrix has equal numbers of rows and columns (order m × m), also called a matrix of order m.
If a matrix has 8 elements, what are all possible orders?
Since mn = 8, possible orders are 1 × 8, 8 × 1, 2 × 4, and 4 × 2.
How can geometric figures be represented as matrices?
Vertices of a polygon are written as columns or rows in a matrix; example: quadrilateral ABCD vertices form a 2 × 4 or 4 × 2 matrix.
Construct a 2 × 2 matrix with element aij = i + j.
Elements: a₁₁ = 2, a₁₂ = 3, a₂₁ = 3, a₂₂ = 4, giving matrix [[2, 3], [3, 4]].
What is the diagonal of a square matrix?
The diagonal of a square matrix consists of elements a₁₁, a₂₂, a₃₃, ..., aₙₙ (from top-left to bottom-right).
Define a matrix. Give one real-world example of how matrices are used in business applications. [2 marks]
State that a matrix is an ordered rectangular array of numbers or functions. Provide one concrete business example: spreadsheets for budgeting, sales projection, or cost estimation with rows and columns for data organisation.
Construct a 3 × 2 matrix A whose elements are given by aij = (3i − 2j)/2, where i = 1, 2, 3 and j = 1, 2. Show all calculations for each element. [5 marks]
Calculate each of the 6 elements using the formula aij = (3i − 2j)/2. For example: a₁₁ = (3·1 − 2·1)/2 = 1/2. Arrange all results in a 3 × 2 rectangular array.
Prove that if a matrix has m × n elements and all possible orders are listed, then the number of ordered pairs (m, n) equals the number of divisors of the total number of elements. Illustrate with a matrix having 12 elements. [6 marks]
Show that if a matrix has k elements (k = m × n), then each divisor d of k gives an order d × (k/d). For k = 12, list all divisors: 1, 2, 3, 4, 6, 12. Each forms a valid order pair: (1,12), (2,6), (3,4), (4,3), (6,2), (12,1) — total 6 pairs corresponding to 6 divisors of 12.
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