MathematicsChapter 2: Matrices and Determinants

Matrix Multiplication

AB is defined when the number of columns of A equals the number of rows of B.

Learning objectives

State the main ideas and notation used in matrix multiplication.
Apply the rules of matrix multiplication in a clear, step-by-step solution.
Check signs, conditions, and the final answer for mathematical accuracy.

Definition

AB is defined when the number of columns of A equals the number of rows of B. If A is m x n and B is n x p, then AB is m x p. The (i,j) entry of AB is row i of A dotted with column j of B. Meaning: you mix a whole row with a whole column. AB is usually not BA, even when both exist.

Where / symbols

MATRIX MULTIPLICATION: AB is defined when the number of columns of A equals the number of rows of B. If A is m x n and B is n x p, then AB is m x p. The (i,j) entry of AB is row i of A dotted with column j of B. Meaning: you mix a whole row with a whole column. AB is usually not BA, even when both exist.
1. SIZE CHECK: Write the sizes in a chain: (2x3)(3x4)=(2x4). (2x3)(2x3) is not defined. I A = A I = A when sizes fit.
2. ROW TIMES COLUMN: Walk along the row and down the column, multiply pairs, add. One entry at a time. Show the sum on its own lines in an exam.

Relationship

A columns = B rows
AB != BA in general

Overview

When it exists, then how to compute:

Size Check

Write the sizes in a chain: (2x3)(3x4)=(2x4). (2x3)(2x3) is not defined. I A = A I = A when sizes fit.

Examples 2x3 times 3x1 = 2x1 · Square: n x n times n x n · BA may be a different size

Row Times Column

Walk along the row and down the column, multiply pairs, add. One entry at a time. Show the sum on its own lines in an exam.

Examples [1 2] times [[3],[4]] = 11 · Work one entry · Do not multiply matching positions like addition

WHAT — MATRIX MULTIPLICATION

RULE — 1. SIZE CHECK

Write the sizes in a chain: (2x3)(3x4)=(2x4). (2x3)(2x3) is not defined. I A = A I = A when sizes fit.

METHOD — 2. ROW TIMES COLUMN

Walk along the row and down the column, multiply pairs, add. One entry at a time. Show the sum on its own lines in an exam.

WORKED EXAMPLE

Worked: [[1,2],[3,4]] [[5,6],[7,8]]. entry 11: 15+27 = 19 entry 12: 16+28 = 22 entry 21: 35+47 = 43 entry 22: 36+48 = 50

How to answer a matrix multiplication question

Identify the given quantities, notation, and required result.
Choose the matching rule and write it before substitution.
Work one algebraic step at a time, keeping brackets and signs clear.
Check restrictions, simplify the result, and verify it where possible.

Worked example

[[1,2],[3,4]] [[5,6],[7,8]].
entry 11: 1*5+2*7 = 19
entry 12: 1*6+2*8 = 22
entry 21: 3*5+4*7 = 43
entry 22: 3*6+4*8 = 50

In simple words

Matrix multiply is not entrywise. That is Hadamard, not this course.

Key points

A columns = B rows
AB != BA in general
inner numbers match
dot product of row and col

Exam preparation

Exam Focus

Quick revision

Revision Zone

Review the key facts, formulas and checks before you move on.