Matrix Multiplication - StudyPulse
StudyPulse Logo
Sign Up
Boost Your VCE Scores Today with StudyPulse
8000+ Questions AI Tutor Help
Create Free Account Learn More
Home Subjects General Mathematics Matrix multiplication

Matrix Multiplication

General Mathematics

About these notes

Area Of Study

Matrices

Unit

Unit 3: General Mathematics Unit 3

Join StudyPulse

Sign up for a free account to access more study resources and track your progress.

Create Free Account Learn More
Get started in 60 seconds
StudyPulse

Matrix Multiplication

General Mathematics
01 May 2026

Matrix Multiplication and Applications

When Can You Multiply?

The product \(AB\) is defined only when the number of columns in \(A\) equals the number of rows in \(B\).

\[A_{m \times k} \times B_{k \times n} = C_{m \times n}\]

How to Multiply

Each entry \(c_{ij}\) in the product \(C\) is the dot product of row \(i\) of \(A\) and column \(j\) of \(B\):

\[c_{ij} = \sum_{r=1}^{k} a_{ir} \cdot b_{rj}\]

Worked Example — \(2 \times 2\) Product

\[A = \begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 0 \\ 2 & 3 \end{pmatrix}\]
\[AB = \begin{pmatrix} 3(5)+1(2) & 3(0)+1(3) \\ 2(5)+4(2) & 2(0)+4(3) \end{pmatrix} = \begin{pmatrix} 17 & 3 \\ 18 & 12 \end{pmatrix}\]

Worked Example — Practical Application

A factory makes two products P and Q. Materials required per unit:

\[\text{Materials} = \begin{pmatrix} 4 & 2 \\ 3 & 5 \end{pmatrix} \quad (\text{rows: P, Q; columns: steel, plastic})\]

Production quantities (units per week):

\[\text{Units} = \begin{pmatrix} 100 \\ 60 \end{pmatrix} \quad (\text{rows: P, Q})\]

Total materials needed:

\[\text{Materials} \times \text{Units} = \begin{pmatrix} 4(100)+2(60) \\ 3(100)+5(60) \end{pmatrix} = \begin{pmatrix} 520 \\ 600 \end{pmatrix}\]

520 kg of steel and 600 kg of plastic are required.

Key Properties

Property Statement
Not commutative \(AB \neq BA\) in general
Associative \((AB)C = A(BC)\)
Identity \(AI = IA = A\)
Zero \(AO = OA = O\)
Distributive \(A(B+C) = AB + AC\)

Powers of a Matrix

For a square matrix \(A\):

\[A^2 = AA, \quad A^3 = AAA, \quad \text{etc.}\]

Powers are used in transition and Markov chain problems to find state distributions after \(n\) steps.

COMMON MISTAKE: Reversing the order of multiplication. If \(A\) is \(2 \times 3\) and \(B\) is \(3 \times 4\), then \(AB\) is valid but \(BA\) requires \(B\) to be \(m \times 2\), which it is not here.

EXAM TIP: Write out the orders of each matrix before multiplying. Then confirm the inner dimensions match and record the outer dimensions for the result order.

Table of Contents