Matrices • Topic 3 of 3

Addition and Subtraction of Matrices, and Multiplication by a Scalar

What is matrix addition and subtraction? When two matrices have the same order, we can add or subtract them by adding or subtracting the corresponding elements (elements in the same position).

Addition: If A = [aᵢⱼ] and B = [bᵢⱼ] are both m × n matrices, then: A + B = [aᵢⱼ + bᵢⱼ]

Subtraction: A − B = [aᵢⱼ − bᵢⱼ]

What is scalar multiplication? Scalar multiplication means multiplying every element of a matrix by the same number (called a scalar). If k is a scalar and A = [aᵢⱼ], then: kA = [k × aᵢⱼ]

Properties of scalar multiplication:

k(A + B) = kA + kB (distributive)
(k + m)A = kA + mA (distributive)
k(mA) = (km)A (associative)
1 × A = A (identity)
0 × A = Zero matrix

Real-life analogy: If matrix A represents the prices of items in two stores, and matrix B represents a price increase, then A + B gives the new prices. Scalar multiplication is like applying a 10% discount (multiply by 0.9) to all prices at once!

┌─────────────────────────────────────────────────────────────┐
│      ADDITION, SUBTRACTION, AND SCALAR MULTIPLICATION        │
└─────────────────────────────────────────────────────────────┘

MATRIX ADDITION (Element by element):

    A = [2  4]    B = [1  3]    A + B = [2+1  4+3] = [3  7]
        [5  7]        [2  6]            [5+2  7+6]   [7  13]
    
    ┌─────┬─────┐   ┌─────┬─────┐   ┌─────┬─────┐
    │ 2   │ 4   │   │ 1   │ 3   │   │ 3   │ 7   │
    ├─────┼─────┤ + ├─────┼─────┤ = ├─────┼─────┤
    │ 5   │ 7   │   │ 2   │ 6   │   │ 7   │ 13  │
    └─────┴─────┘   └─────┴─────┘   └─────┴─────┘


MATRIX SUBTRACTION:

    A = [7  5]    B = [3  2]    A - B = [7-3  5-2] = [4  3]
        [6  8]        [1  4]            [6-1  8-4]   [5  4]


SCALAR MULTIPLICATION:

    A = [1  2]    3A = [3×1  3×2] = [3  6]
        [3  4]          [3×3  3×4]   [9  12]
    
    Every element gets multiplied by the scalar!


COMBINED OPERATIONS (BODMAS rules apply):

    Given A = [1  2], B = [2  3], find 2A + 3B
              [3  4]     [4  5]
    
    Step 1: 2A = [2  4]
                [6  8]
    
    Step 2: 3B = [6  9]
                [12 15]
    
    Step 3: 2A + 3B = [2+6  4+9] = [8  13]
                     [6+12 8+15]   [18 23]


RULES AT A GLANCE:

┌─────────────────────────────────────────────────────────┐
│  Operation        │  Condition        │  How to Compute  │
├───────────────────┼───────────────────┼──────────────────┤
│  A + B            │  Same order       │  Add elements    │
│  A - B            │  Same order       │  Subtract elements│
│  kA               │  Any matrix       │  Multiply each   │
│                   │                   │  element by k    │
└───────────────────┴───────────────────┴──────────────────┘

Solved Examples

Worked Example

If A = [1 2; 3 4] and B = [5 6; 7 8], find A + B and A − B.

Solution

Step 1: A + B = [1+5, 2+6; 3+7, 4+8] = [6, 8; 10, 12]
Step 2: A − B = [1−5, 2−6; 3−7, 4−8] = [−4, −4; −4, −4]

Answer: A + B = [6 8; 10 12]; A − B = [−4 −4; −4 −4]

Worked Example

Given A = [2 4; 6 8], find 5A.

Solution

Step 1: Multiply each element by 5: 5×2 = 10, 5×4 = 20, 5×6 = 30, 5×8 = 40

Answer: 5A = [10 20; 30 40]

Worked Example

If A = [3 1; 2 4] and B = [2 3; 1 5], find 3A − 2B.

Solution

Step 1: 3A = [9 3; 6 12]
Step 2: 2B = [4 6; 2 10]
Step 3: 3A − 2B = [9−4, 3−6; 6−2, 12−10] = [5, −3; 4, 2]

Answer: 3A − 2B = [5 −3; 4 2]

Key Points

Addition/subtraction only possible for matrices of same order
Add/subtract corresponding elements (same row, same column)
Scalar multiplication multiplies EVERY element by the scalar
Scalar multiplication is distributive over addition
Zero matrix (all zeros) is the additive identity
Order remains the same after addition, subtraction, and scalar multiplication

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.Matrices are added:

Explanation: Add corresponding entries.

Q2.Scalar multiplication multiplies:

Explanation: Each entry by the scalar.

Q3.$2\begin{bmatrix}1&2\\3&4\end{bmatrix}=$

Explanation: Double each entry.

Q4.$A-A=$

Explanation: All entries become $0$.

Practice more MCQs → 📝 Assignment · 35 marks