# Multiplying Matrices

How to multiply matrices: properties, examples, and their solutions.

## Example 1

Multiply the elements in the following order.

Row 1, Column 1:

(Row 1, →) × (Column 1, ↓)

Row 1, Column 2:

(Row 1, →) × (Column 2, ↓)

Row 2, Column 1:

(Row 2, →) × (Column 1, ↓)

Row 2, Column 2:

(Row 2, →) × (Column 2, ↓)

To multiply matrices, this condition is needed:

(number of former matrix's row) = (number of latter matrix's column)

## Example 2

Row 1, Column 1:

(Row 1, →) × (Column 1, ↓)

Row 1, Column 2:

(Row 1, →) × (Column 2, ↓)

Row 2, Column 1:

(Row 2, →) × (Column 1, ↓)

Row 2, Column 2:

(Row 2, →) × (Column 2, ↓)

For multiplying matrices,*AB* ≠ *BA*.

(Check the previous example's answer.)

## Identity Matrix

The identity matrix is a matrix

whose diagonal elements are 1

and the other elements are 0.

It's denoted by *I*.

*AI* = *IA* = *A*

The order doesn't matter.

So *I*^{2} = *II* = *I*.

## Example 3

*AI* = *A*

So *AIA* = *AA*.

Row 1, Column 1:

(Row 1, →) × (Column 1, ↓)

Row 1, Column 2:

(Row 1, →) × (Column 2, ↓)

Row 2, Column 1:

(Row 2, →) × (Column 1, ↓)

Row 2, Column 2:

(Row 2, →) × (Column 2, ↓)

## Zero Matrix

The zero matrix is a matrix

whose elements are all 0.

It's denoted by *O*.*AO* = *OA* = *O*

But *AB* = *O* doesn't always mean*A* = *O* or *B* = *O*.

Even if *A* ≠ *O* and *B* ≠ *O*,*AB* can be *O*.

(See the next example.)

## Example 4

Recall that a counterexample is an example

that makes the given statement false.

And a conditional statement is false

if the hypothesis is true and the conclusion is false.

Conditional statement: truth value

So find a counterexample

that makes *AB* = *O* and *BA* ≠ *O*.

Assume *A* and *B*

that seems to be the counterexample.*A* = [1 0 / 2 0], *B* = [0 0 / 0 1]

Show *AB* = *O*.

(true hypothesis)

Show *BA* ≠ *O*.

(false conclusion)

'*A* = [1 0 / 2 0], *B* = [0 0 / 0 1]' makes

the hypothesis (*AB* = *O*) true

and the conclusion (*BA* = *O*) false.

So the counterexample is

'*A* = [1 0 / 2 0], *B* = [0 0 / 0 1]'.