Matrix Multiplication

Key Terms

Matrix product AB: Defined only when the number of columns of A equals the number of rows of B.
Dimensions: An (m × n) matrix times an (n × p) matrix gives an (m × p) result.
Entry (i, j) of AB: The dot product of row i of A with column j of B.
Non-commutativity: In general AB ≠ BA; order of multiplication matters.
Identity: AI = IA = A for any matrix A compatible with the identity matrix I.
Associativity: (AB)C = A(BC) always holds; distributivity: A(B + C) = AB + AC.

Matrix Multiplication

When is AB defined? A must be m×n and B must be n×p.
(The number of columns in A must equal the number of rows in B.)
The result AB has order m×p.

Element formula: c_ij = (row i of A) · (column j of B) = ∑ a_ik b_kj

Key Properties

Property	Statement	Note
NOT commutative	AB ≠ BA (in general)	Order matters!
Associative	(AB)C = A(BC)	Always true
Distributive	A(B + C) = AB + AC	Always true
Identity	AI = IA = A	I is the identity matrix
Zero product	AB = O does NOT imply A = O or B = O	Matrices are unusual here

Worked Example 1 — 2×2 by 2×2 Multiplication

Find AB where A =

3	1
2	4

and B =

2	−1
0	3

A is 2×2, B is 2×2 → columns of A (2) = rows of B (2) ✓ → result is 2×2.

Calculate each element:

Position	Calculation	Result
c₁₁	Row 1 of A · Col 1 of B = (3)(2) + (1)(0)	6 + 0 = 6
c₁₂	Row 1 of A · Col 2 of B = (3)(−1) + (1)(3)	−3 + 3 = 0
c₂₁	Row 2 of A · Col 1 of B = (2)(2) + (4)(0)	4 + 0 = 4
c₂₂	Row 2 of A · Col 2 of B = (2)(−1) + (4)(3)	−2 + 12 = 10

AB =

6	0
4	10

Worked Example 2 — 2×3 by 3×2

Find AB where A =

1	2	0
3	−1	4

and B =

2	5
1	−3
4	0

A is 2×3, B is 3×2 → columns of A (3) = rows of B (3) ✓ → result AB is 2×2.

c₁₁ = (1)(2) + (2)(1) + (0)(4) = 2 + 2 + 0 = 4

c₁₂ = (1)(5) + (2)(−3) + (0)(0) = 5 − 6 + 0 = −1

c₂₁ = (3)(2) + (−1)(1) + (4)(4) = 6 − 1 + 16 = 21

c₂₂ = (3)(5) + (−1)(−3) + (4)(0) = 15 + 3 + 0 = 18

AB =

4	−1
21	18

Hot Tip: Matrix multiplication is NOT commutative — AB ≠ BA in general. Also, even if AB is defined, BA may not be defined (if the dimensions don’t match). Always write matrix products in the specified order.

Full Lesson: Matrix Multiplication

Why Multiply Matrices This Way?

Matrix multiplication is designed to combine transformations. When you multiply a price row matrix by a quantity matrix, you get total costs. When you multiply transformation matrices in geometry, you combine multiple transformations into one. The row-by-column method is precisely what makes this work.

The Compatibility Rule

Before computing AB, check:

Write out the orders: A is (m × n), B is (n × p)
The inner dimensions must match (both must be n)
The outer dimensions give the result order: m × p

Quick check method: Write the orders side by side: (2×3)(3×4). The middle two numbers must match (3 = 3). The result order is the outer two numbers: 2×4.

Computing Each Element

Element c_ij in the product AB is found by:

Taking row i from matrix A
Taking column j from matrix B
Multiplying corresponding entries and summing the results

This is called the dot product of a row and a column.

Why AB ≠ BA?

In ordinary numbers, 3 × 5 = 5 × 3. But for matrices, the row-column pairing changes completely when you swap the order. Even when both products are defined (e.g., two 2×2 matrices), the results are usually different. This is one of the most important differences between matrix algebra and ordinary algebra.

The Identity Matrix

The identity matrix I behaves like the number 1: AI = IA = A for any matrix A (provided dimensions are compatible). This is why I is called the “multiplicative identity” for matrices.

Pricing Application: If prices P = [5, 8, 3] (a 1×3 row matrix) and quantities Q is a 3×2 matrix (quantities of each item for two orders), then PQ gives a 1×2 row matrix of total costs for each order.

Lesson Tip: When computing matrix products by hand, write out the row and column as a list, pair them up, multiply each pair, then add. Work systematically through each element — left to right along each row, top to bottom for each column. A common error is mixing up row and column indices.

Mastery Practice

See Answers ➔

Fluency
For each matrix product below, state whether it is defined and, if so, give the order of the result.
1. A is 2×3, B is 3×4. Is AB defined? What is its order?
2. A is 2×2, B is 3×2. Is AB defined?
3. A is 4×1, B is 1×3. Is AB defined? What is its order?
4. A is 3×3, B is 3×1. Is AB defined? What is its order?
Fluency
Calculate AB where A =

2 1

3 4

and B =

1 0

2 −1

Show all intermediate calculations.
Fluency
Calculate AB where A =

3 −1

0 2

and B =

−1 4

2 1
Fluency
Let A =

5 2

−1 3

and I =

1 0

0 1

.

Calculate AI and IA and verify both equal A.
Understanding
Let A =

1 2

3 4

and B =

2 −1

0 3

.

Calculate AB and BA and show that AB ≠ BA.
Understanding
Let A =

1 2

0 1

, B =

2 0

1 1

, C =

1 3

2 0

.

Calculate (AB)C and A(BC) and verify that they are equal (associative property).

Understanding

A shop sells three products. The unit prices (in dollars) are given by row matrix P = [4.50, 3.00, 6.00]. The quantity matrix for two orders is:

Q =

20	15
30	25
10	20

Rows = products 1, 2, 3. Columns = Orders A and B.

State the orders of P and Q. Is PQ defined? What will the order of PQ be?
Calculate PQ.
Interpret each entry in PQ in the context of this problem.

Understanding
A message is encoded using the matrix E =

2 5

1 3

. The original message vector is m =

3

1

.
1. Calculate the encoded message Em.
2. The inverse of E is E⁻¹ =
  
  3 −5
  
  −1 2
  
  . Verify that E⁻¹(Em) = m.
Problem Solving
Let A =

2 1

k 3

.
1. Calculate A² = A × A in terms of k.
2. Recall that det(A²) = (det A)². Find det(A) in terms of k.
3. Find the value(s) of k such that det(A²) = 25.

Problem Solving

A manufacturer uses two types of parts to make two products. The resources (machine-hours and labour-hours) needed per part are given in matrix R:

R =

2	1
3	0
1	2

(rows = resource types: machine-hrs, labour-hrs, quality-hrs; columns = Part 1, Part 2)

The parts required per product are given in matrix P:

P =

4	2
1	3

(rows = Part 1, Part 2; columns = Product A, Product B)

State the orders of R and P. Is RP defined? What is the order of RP?
Calculate RP.
Interpret the element in row 1, column 2 of RP in context.

3	−5
−1	2