Matrices — Part 2: Multiplication of matrices: visual lesson, ICSE Class 10 Mathematics

Lesson

Matrices, Part 2

Multiplying matrices: rows of the first by columns of the second.

In this lesson you will

Decide whether two matrices can be multiplied and state the order of the product
Multiply a 2 × 2 matrix by a 2 × 2 or 2 × 1 matrix, and a 1 × 2 matrix by a 2 × 2 matrix
Show with an example that AB ≠ BA, and use AI = IA = A
Find unknowns from matrix products and work with A²

Compatibility for multiplication.

Concept

What you will be able to do

Check compatibility for multiplication and state the order of a product
Use the row-by-column rule
Show that $AB \ne BA$ in general, and use $AI = IA = A$
Find unknowns from a product
Work out $A^2$ and expressions such as $A^2 - 3A + 5I$

Concept

When is a product defined?

$AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$ . If $A$ is $m \times n$ and $B$ is $n \times p$ , then $AB$ is $m \times p$ .

Inner numbers must match; the outer numbers give the order.
First matrix	Second matrix	Product defined?	Order of product
$2 \times 2$	$2 \times 2$	yes	$2 \times 2$
$2 \times 2$	$2 \times 1$	yes	$2 \times 1$
$1 \times 2$	$2 \times 2$	yes	$1 \times 2$
$2 \times 1$	$2 \times 2$	no	not defined
$2 \times 2$	$1 \times 2$	no	not defined

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