Matrix Addition, Subtraction and Scalar Multiplication

Key Terms

Matrix addition/subtraction: Add or subtract corresponding entries; only defined for matrices of the SAME order.
Scalar multiplication: Multiply every entry by the scalar k; (kA)_ij = k × a_ij.
Transpose A^T: Swap rows and columns; entry (i, j) of A becomes entry (j, i) of A^T.
Commutative law: A + B = B + A (addition commutes); but AB ≠ BA in general (multiplication does not).
Conformable for addition: Two matrices must have identical order (same number of rows AND columns) to be added.
Notation: Capital letters for matrices (A, B, C); lower case for scalars (k, c); boldface optional.

Matrix Operations

Addition & Subtraction: Only defined for matrices of the same order.
Add or subtract corresponding elements (same position).

Scalar Multiplication: Multiply every element in the matrix by the scalar k.

Key Properties

Property	Statement
Commutative (addition)	A + B = B + A
Associative (addition)	(A + B) + C = A + (B + C)
Distributive (scalar)	k(A + B) = kA + kB
Scalar distributive	(j + k)A = jA + kA
Zero matrix	A + O = A (O is the zero matrix)

Worked Example 1 — Matrix Addition

Calculate A + B where A =

3	-1	5
2	4	-2

and B =

1	6	-3
-5	0	7

Both are 2 × 3 → addition is defined. Add element by element:

A + B =

3+1	−1+6	5+(−3)
2+(−5)	4+0	−2+7

4	5	2
−3	4	5

Worked Example 2 — Scalar Multiplication and Subtraction

Find 3A − 2B where A =

2	−1
0	4

and B =

1	3
−2	1

Step 1: Calculate 3A — multiply every element of A by 3:

3A =

6	−3
0	12

Step 2: Calculate 2B — multiply every element of B by 2:

2B =

2	6
−4	2

Step 3: Subtract element by element:

3A − 2B =

6−2	−3−6
0−(−4)	12−2

4	−9
4	10

Hot Tip: You CANNOT add or subtract matrices of different orders — always check that both matrices have the same number of rows AND the same number of columns before attempting the operation. If the orders differ, the operation is undefined.

Full Lesson: Matrix Operations

Addition and Subtraction

Matrix addition and subtraction are only defined when both matrices have exactly the same order. We operate on corresponding elements — the element in position (i, j) of the result comes from adding or subtracting the elements at position (i, j) from each matrix.

Why same order? Think of it like adding apples to apples. Each element in a matrix represents a specific piece of data (e.g., stock of Product A in Warehouse 2). You can only meaningfully add data that represents the same thing. A 2×3 matrix and a 3×2 matrix both have 6 elements but store fundamentally different structures.

Scalar Multiplication

Multiplying a matrix by a scalar (a single number) multiplies every element in the matrix by that number. The order of the matrix does not change.

Example: If prices go up by 10%, multiply the price matrix by 1.1. If you want to double a matrix, multiply by 2. If you want the negative of a matrix, multiply by −1.

Combined Operations

Expressions like 3A − 2B follow the same order of operations as regular algebra:

First compute the scalar multiples (3A and 2B)
Then perform the subtraction element by element

Finding an Unknown Matrix

If X + A = B, we can rearrange just like algebra: X = B − A. We can add, subtract, and scalar multiply matrices on both sides of an equation.

Example: Find X if 2X − A = B.
2X = B + A → X = ½(B + A)
This means: add B + A element by element, then divide every element by 2.

Real-World Application: Tracking Data Over Time

Matrices are ideal for tracking quantities that change over time. Stock levels, sports scores, financial data, temperature readings — all can be stored in matrices and updated using addition and subtraction.

Lesson Tip: When solving for an unknown matrix, isolate the unknown matrix on one side using the same algebraic techniques you know (add to both sides, divide both sides by a scalar). The only new rule is: always check that orders match before adding or subtracting.

Mastery Practice

See Answers ➔

Fluency
Calculate A + B where A =

5 −3

2 7

and B =

4 1

−6 3

Fluency

Calculate A − B where A =

8	3	−1
5	−2	6

and B =

2	−4	3
−1	5	2

Fluency
Let A =

3 −2

1 4

and B =

−1 3

2 −1
1. Calculate 3A.
2. Calculate −2B.
3. Hence calculate 3A − 2B.
Fluency
Find matrix X given that X + A = B, where A =

2 5

−1 3

and B =

7 2

4 −1
Understanding
Let A =

1 2

3 4

, B =

5 −1

0 2

, C =

2 3

−1 1

Calculate:
1. A + B
2. 2B − C
3. A + B + C
4. Verify that A + B = B + A (calculate B + A and confirm it equals your answer from part (a)).
Understanding
A retailer tracks stock levels using matrices. Stock at Monday open M, deliveries D, and Wednesday sales S are:

M =

20 15

30 10

, D =

8 12

5 15

, S =

14 11

18 6

Rows = Items A and B. Columns = Shop 1 and Shop 2.
1. Calculate the stock after Tuesday deliveries: M + D.
2. Calculate closing stock after Wednesday sales: (M + D) − S.
3. Does any stock level drop below zero after sales? What does this mean?
Understanding
Find the values of a and b:

2

a 3

1 b

−

4 −1

3 5

=

0 7

−1 3
Understanding
Temperature readings (in °C) for three days across two cities are stored in matrix T:

T =

28 32

25 30

31 27

Rows = Days 1, 2, 3. Columns = City 1, City 2. After reviewing the data, it is found that each thermometer was calibrated 2°C too high. Write the corrected temperature matrix and explain how you obtained it.
Problem Solving
A household records monthly spending in two budget categories across two months:

January: J =

$800 $450

$300 $200

February: F =

$750 $500

$280 $220

Rows = Food, Rent. Columns = Actual, Budgeted.
1. Find the total 2-month spending matrix J + F.
2. Find the monthly average spending matrix ½(J + F).
3. Find the difference F − J. Which categories increased from January to February in terms of actual spending (column 1)?
Problem Solving
Let A =

2 4

1 3

and B =

6 12

3 9
1. Find all values of k such that kA = B.
2. Is there a value of k such that kA = B + A? Find it.
3. What relationship between A and B do you notice? Express B in terms of A.

4	1
−6	3

$800	$450
$300	$200

$750	$500
$280	$220