The Central Idea

Suppose

\[ A= \begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix} \]

and

\[ B= \begin{bmatrix} 5 & 6\\ 7 & 8 \end{bmatrix}. \]

Addition is entrywise:

\[ A+B= \begin{bmatrix} 1+5 & 2+6\\ 3+7 & 4+8 \end{bmatrix} = \boxed{ \begin{bmatrix} 6 & 8\\ 10 & 12 \end{bmatrix} }. \]

But multiplication is not entrywise.

Instead,

\[ AB= \begin{bmatrix} 1(5)+2(7) & 1(6)+2(8)\\ 3(5)+4(7) & 3(6)+4(8) \end{bmatrix} = \boxed{ \begin{bmatrix} 19 & 22\\ 43 & 50 \end{bmatrix} }. \]

Matrix multiplication combines row-column structure rather than matching positions directly.

Matrix Addition

Two matrices can be added only when they have the same dimensions.

If

\[ A=[a_{ij}] \]

and

\[ B=[b_{ij}], \]

then

\[ \boxed{ A+B=[a_{ij}+b_{ij}] }. \]

Example: Matrix Addition

Let

\[ A= \begin{bmatrix} 2 & -1\\ 5 & 3 \end{bmatrix} \]

and

\[ B= \begin{bmatrix} 4 & 6\\ -2 & 7 \end{bmatrix}. \]

Then

\[ A+B= \begin{bmatrix} 6 & 5\\ 3 & 10 \end{bmatrix}. \]

Why Dimensions Must Match for Addition

Addition pairs corresponding entries.

A

\[ 2\times3 \]

matrix has six positions arranged differently from a

\[ 3\times2 \]

matrix.

Because their entry positions do not correspond, their sum is undefined.

Matrix Subtraction

Matrix subtraction is also entrywise:

\[ \boxed{ A-B=[a_{ij}-b_{ij}] }. \]

The matrices must again have the same dimensions.

Example: Matrix Subtraction

Let

\[ A= \begin{bmatrix} 7 & 3\\ 4 & 9 \end{bmatrix}, \qquad B= \begin{bmatrix} 2 & 5\\ 1 & 6 \end{bmatrix}. \]

Then

\[ A-B= \boxed{ \begin{bmatrix} 5 & -2\\ 3 & 3 \end{bmatrix} }. \]

Scalar Multiplication

A scalar multiplies every entry of a matrix.

If

\[ k\in\mathbb{R}, \]

then

\[ \boxed{ kA=[ka_{ij}] }. \]

Example: Scalar Multiplication

For

\[ A= \begin{bmatrix} 2 & -1\\ 3 & 5 \end{bmatrix}, \]

we have

\[ 3A= \boxed{ \begin{bmatrix} 6 & -3\\ 9 & 15 \end{bmatrix} }. \]

Negative Scalars

Multiplying by

\[ -1 \]

changes the sign of every entry:

\[ -A= \begin{bmatrix} -a_{11} & -a_{12}\\ -a_{21} & -a_{22} \end{bmatrix}. \]

This allows subtraction to be written as

\[ A-B=A+(-B). \]

The Zero Matrix in Addition

The zero matrix acts as the additive identity.

For any matrix \(A\) of matching dimensions,

\[ \boxed{ A+0=A }. \]

Additive Inverses

Every matrix \(A\) has an additive inverse

\[ -A. \]

Then

\[ \boxed{ A+(-A)=0 }. \]

Basic Addition Properties

For matrices of compatible dimensions:

\[ A+B=B+A, \] \[ (A+B)+C=A+(B+C), \] \[ A+0=A. \]

Matrix addition behaves much like vector addition.

Scalar Distribution

Scalar multiplication distributes over matrix addition:

\[ \boxed{ k(A+B)=kA+kB }. \]

Also,

\[ (k+\ell)A=kA+\ell A. \]

Matrix Multiplication Is Different

Matrix multiplication is not entrywise multiplication.

It is defined through row-column products.

This operation is designed so that matrices represent compositions of linear relationships.

Dimension Rule for Multiplication

If

\[ A \]

is

\[ m\times n \]

and

\[ B \]

is

\[ n\times p, \]

then

\[ AB \]

is defined and has dimensions

\[ \boxed{m\times p}. \]

The Inner Dimensions Must Match

For

\[ (m\times n)(n\times p), \]

the inner dimensions

\[ n \]

must match.

The outer dimensions

\[ m \]

and

\[ p \]

become the dimensions of the product.

Dimension Example

If

\[ A \]

is

\[ 2\times3 \]

and

\[ B \]

is

\[ 3\times4, \]

then

\[ AB \]

is defined and has dimensions

\[ \boxed{2\times4}. \]

Undefined Product Example

If

\[ A \]

is

\[ 2\times3 \]

and

\[ B \]

is

\[ 2\times4, \]

then

\[ AB \]

is not defined because the inner dimensions

\[ 3 \]

and

\[ 2 \]

do not match.

Row by Column

Suppose

\[ A= \begin{bmatrix} a & b \end{bmatrix} \]

and

\[ B= \begin{bmatrix} c\\ d \end{bmatrix}. \]

Then

\[ AB= \boxed{ ac+bd }. \]

A row times a column produces a scalar.

The Entry Formula

If

\[ C=AB, \]

then the entry in row \(i\), column \(j\) is

\[ \boxed{ c_{ij} = \sum_{k=1}^{n}a_{ik}b_{kj} }. \]

In words: take row \(i\) of \(A\), column \(j\) of \(B\), multiply corresponding entries, and add.

Two-by-Two Multiplication

Let

\[ A= \begin{bmatrix} a & b\\ c & d \end{bmatrix}, \qquad B= \begin{bmatrix} e & f\\ g & h \end{bmatrix}. \]

Then

\[ \boxed{ AB= \begin{bmatrix} ae+bg & af+bh\\ ce+dg & cf+dh \end{bmatrix} }. \]

Worked Multiplication Example

Let

\[ A= \begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}, \qquad B= \begin{bmatrix} 5 & 6\\ 7 & 8 \end{bmatrix}. \]

The first entry is

\[ 1(5)+2(7)=19. \]

The upper-right entry is

\[ 1(6)+2(8)=22. \]

Finish the Product

The lower-left entry is

\[ 3(5)+4(7)=43. \]

The lower-right entry is

\[ 3(6)+4(8)=50. \]

Therefore,

\[ \boxed{ AB= \begin{bmatrix} 19 & 22\\ 43 & 50 \end{bmatrix} }. \]

Rectangular Multiplication

Let

\[ A= \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6 \end{bmatrix} \]

and

\[ B= \begin{bmatrix} 1 & 0\\ 0 & 1\\ 1 & 1 \end{bmatrix}. \]

The product is defined because

\[ (2\times3)(3\times2). \]

The result will be

\[ 2\times2. \]

Compute the Rectangular Product

First row, first column:

\[ 1(1)+2(0)+3(1)=4. \]

First row, second column:

\[ 1(0)+2(1)+3(1)=5. \]

Second row, first column:

\[ 4(1)+5(0)+6(1)=10. \]

Second row, second column:

\[ 4(0)+5(1)+6(1)=11. \]

Thus,

\[ \boxed{ AB= \begin{bmatrix} 4 & 5\\ 10 & 11 \end{bmatrix} }. \]

Matrix–Vector Multiplication

A matrix can multiply a column vector.

Let

\[ A= \begin{bmatrix} 2 & 3\\ 4 & -1 \end{bmatrix} \]

and

\[ \mathbf{x}= \begin{bmatrix} x\\ y \end{bmatrix}. \]

Then

\[ A\mathbf{x} = \boxed{ \begin{bmatrix} 2x+3y\\ 4x-y \end{bmatrix} }. \]

Why Matrix Multiplication Represents Systems

The equation

\[ A\mathbf{x}=\mathbf{b} \]

with

\[ A= \begin{bmatrix} 2 & 3\\ 4 & -1 \end{bmatrix}, \qquad \mathbf{b}= \begin{bmatrix} 7\\ 5 \end{bmatrix} \]

means

\[ \begin{cases} 2x+3y=7,\\ 4x-y=5. \end{cases} \]

Matrix multiplication reproduces the linear combinations in the system exactly.

Rows as Linear Combinations

Each entry of

\[ A\mathbf{x} \]

is the dot product of one row of \(A\) with the vector \(\mathbf{x}\).

Thus, each row computes one linear combination of the variables.

Columns as Contributions

Matrix multiplication can also be interpreted through columns.

If

\[ A= \begin{bmatrix} \vert & \vert\\ \mathbf{a}_1 & \mathbf{a}_2\\ \vert & \vert \end{bmatrix}, \]

then

\[ A \begin{bmatrix} x_1\\ x_2 \end{bmatrix} = \boxed{ x_1\mathbf{a}_1+x_2\mathbf{a}_2 }. \]

The output is a linear combination of the columns of \(A\).

Example: Column Combination

Let

\[ A= \begin{bmatrix} 1 & 3\\ 2 & 4 \end{bmatrix}, \qquad \mathbf{x}= \begin{bmatrix} 5\\ -1 \end{bmatrix}. \]

Then

\[ A\mathbf{x} = 5 \begin{bmatrix} 1\\ 2 \end{bmatrix} - \begin{bmatrix} 3\\ 4 \end{bmatrix} = \boxed{ \begin{bmatrix} 2\\ 6 \end{bmatrix} }. \]

Matrix Multiplication and Transformations

A matrix can represent a transformation of vectors.

For example,

\[ A= \begin{bmatrix} 2 & 0\\ 0 & 3 \end{bmatrix} \]

maps

\[ \begin{bmatrix} x\\ y \end{bmatrix} \]

to

\[ \begin{bmatrix} 2x\\ 3y \end{bmatrix}. \]

This scales the \(x\)-coordinate by \(2\) and the \(y\)-coordinate by \(3\).

Composition of Transformations

If matrix \(B\) transforms a vector first and matrix \(A\) acts afterward, then the combined transformation is

\[ \boxed{AB}. \]

Order matters because the rightmost transformation acts first.

Why Multiplication Order Matters

Matrix multiplication is generally not commutative:

\[ \boxed{ AB\ne BA } \]

in general.

This differs from ordinary real-number multiplication.

Example of Noncommutativity

Let

\[ A= \begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix}, \qquad B= \begin{bmatrix} 2 & 0\\ 0 & 1 \end{bmatrix}. \]

Then

\[ AB= \begin{bmatrix} 2 & 1\\ 0 & 1 \end{bmatrix}. \]

But

\[ BA= \begin{bmatrix} 2 & 2\\ 0 & 1 \end{bmatrix}. \]

Therefore,

\[ \boxed{AB\ne BA}. \]

Sometimes \(BA\) Is Not Even Defined

If

\[ A \]

is

\[ 2\times3 \]

and

\[ B \]

is

\[ 3\times4, \]

then

\[ AB \]

is defined.

But

\[ BA \]

would require multiplying

\[ (3\times4)(2\times3), \]

which is undefined because

\[ 4\ne2. \]

The Identity Matrix

The identity matrix acts like multiplicative \(1\).

For a compatible matrix \(A\),

\[ \boxed{ IA=A } \]

and

\[ \boxed{ AI=A }. \]

Two-by-Two Identity

\[ I_2= \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}. \]

For any

\[ 2\times2 \]

matrix \(A\),

\[ I_2A=A \]

and

\[ AI_2=A. \]

Zero Matrix in Multiplication

For compatible dimensions,

\[ A0=0 \]

and

\[ 0A=0. \]

The zero matrix absorbs multiplication much like the number \(0\).

Associativity of Matrix Multiplication

When dimensions are compatible,

\[ \boxed{ (AB)C=A(BC) }. \]

Thus, a chain of transformations can be grouped in different ways without changing the final result.

Distributive Properties

Matrix multiplication distributes over addition:

\[ \boxed{ A(B+C)=AB+AC } \]

and

\[ \boxed{ (A+B)C=AC+BC }. \]

Scalar Compatibility

For a scalar \(k\),

\[ \boxed{ k(AB)=(kA)B=A(kB) }. \]

Scalars can be moved through a matrix product in this way.

Matrix Powers

If \(A\) is square, then powers such as

\[ A^2 \]

and

\[ A^3 \]

are defined by repeated matrix multiplication:

\[ A^2=AA, \] \[ A^3=AAA. \]

Matrix Powers Require Square Matrices

For ordinary powers such as

\[ A^2, \]

the product

\[ AA \]

must be defined.

This requires \(A\) to be square.

Example: Matrix Square

Let

\[ A= \begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix}. \]

Then

\[ A^2= \begin{bmatrix} 1 & 2\\ 0 & 1 \end{bmatrix}. \]

Do Not Square Entries Individually

Matrix squaring means

\[ A^2=AA, \]

not

\[ [a_{ij}^2]. \]

Entrywise squaring is a different operation.

Transpose and Addition

Transpose distributes over addition:

\[ \boxed{ (A+B)^T=A^T+B^T }. \]

Transpose and Multiplication

The transpose of a product reverses the order:

\[ \boxed{ (AB)^T=B^TA^T }. \]

This reversal mirrors the importance of order in matrix multiplication.

Why the Order Reverses

The rows and columns used in \(AB\) switch roles under transpose.

Reversing the order preserves the required row-column pairings.

Data Example

Suppose

\[ Q= \begin{bmatrix} 3 & 2\\ 5 & 4 \end{bmatrix} \]

records quantities of two products purchased by two stores.

Suppose the price vector is

\[ \mathbf{p}= \begin{bmatrix} 10\\ 6 \end{bmatrix}. \]

Then

\[ Q\mathbf{p} \]

computes each store's total cost.

Compute the Data Product

\[ Q\mathbf{p} = \begin{bmatrix} 3(10)+2(6)\\ 5(10)+4(6) \end{bmatrix} = \boxed{ \begin{bmatrix} 42\\ 74 \end{bmatrix} }. \]

The first store spends \$42 and the second \$74.

Units in Matrix Multiplication

In the previous example:

Matrix multiplication can therefore preserve meaningful unit structure.

Weighted Sums

Matrix multiplication naturally computes weighted sums.

If

\[ \mathbf{w}= \begin{bmatrix} w_1\\ w_2\\ \vdots\\ w_n \end{bmatrix}, \]

then each row of

\[ A\mathbf{w} \]

forms a weighted combination of that row's entries.

Networks and Transition Matrices Preview

Repeated multiplication by a matrix can model movement or change through a network or state system.

For example,

\[ \mathbf{x}_{n+1}=A\mathbf{x}_n \]

describes a state updated by the same matrix at each step.

This later connects to recurrences, Markov chains, dynamical systems, and eigenvalues.

Matrix Products Can Encode Multi-Stage Processes

If

\[ B \]

converts raw inputs to intermediate quantities and

\[ A \]

converts those intermediates to outputs, then

\[ AB \]

represents the combined process.

This is why matrix multiplication is built around composition rather than entrywise matching.

Dimension Checks Should Come First

Before multiplying matrices:

  1. state the dimensions of each matrix;
  2. check that the inner dimensions match;
  3. predict the dimensions of the result.

This catches many errors before any arithmetic begins.

Predict the Product Shape

If

\[ A \]

is

\[ 4\times2 \]

and

\[ B \]

is

\[ 2\times5, \]

then

\[ AB \]

must be

\[ \boxed{4\times5}. \]

If your computed product has another shape, something is wrong.

A Reliable Addition Workflow

  1. Check that the dimensions match.
  2. Add corresponding entries.
  3. Keep the same matrix dimensions.
  4. Interpret the result in context if the matrices represent data.

A Reliable Scalar-Multiplication Workflow

  1. Identify the scalar.
  2. Multiply every entry by that scalar.
  3. Keep the matrix dimensions unchanged.
  4. Check signs and units.

A Reliable Matrix-Multiplication Workflow

  1. Write the dimensions of both matrices.
  2. Verify that the inner dimensions match.
  3. Predict the product dimensions.
  4. Take one row of the first matrix and one column of the second.
  5. Multiply corresponding entries and add.
  6. Place the result in the matching row-column position.
  7. Repeat for every output entry.

A Reliable Matrix–Vector Workflow

  1. Check that the vector length equals the number of matrix columns.
  2. Take each matrix row dot product with the vector.
  3. Place each result into the corresponding output entry.
  4. Interpret the output as a transformed vector or collection of linear combinations.

Common Errors

Adding Matrices of Different Dimensions

Matrix addition requires identical dimensions.

Multiplying Matrices Entrywise

Ordinary matrix multiplication uses row-column products, not corresponding-entry products.

Checking the Wrong Dimensions

For \(AB\), compare the columns of \(A\) with the rows of \(B\).

Predicting the Wrong Product Dimensions

The result uses the outer dimensions.

Assuming \(AB=BA\)

Matrix multiplication is generally not commutative.

Assuming \(BA\) Exists Whenever \(AB\) Exists

The reverse product may have incompatible dimensions.

Forgetting All Terms in a Row-Column Product

Every corresponding pair must be multiplied and included in the sum.

Squaring Entries Instead of Multiplying the Matrix by Itself

\(A^2\) means \(AA\).

Ignoring Units

In applied matrix products, row-column combinations should have meaningful units.

Changing Matrix Order During Composition

In \(AB\mathbf{x}\), the rightmost transformation acts first.

Practice

1. Matrix Addition

Compute

\[ \begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6\\ 7 & 8 \end{bmatrix}. \]
Check Answer \[ \boxed{ \begin{bmatrix} 6 & 8\\ 10 & 12 \end{bmatrix} }. \]

2. Matrix Subtraction

Compute

\[ \begin{bmatrix} 7 & 5\\ 4 & 9 \end{bmatrix} - \begin{bmatrix} 2 & 3\\ 1 & 6 \end{bmatrix}. \]
Check Answer \[ \boxed{ \begin{bmatrix} 5 & 2\\ 3 & 3 \end{bmatrix} }. \]

3. Scalar Multiplication

Compute

\[ -2 \begin{bmatrix} 3 & -1\\ 4 & 5 \end{bmatrix}. \]
Check Answer \[ \boxed{ \begin{bmatrix} -6 & 2\\ -8 & -10 \end{bmatrix} }. \]

4. Check Multiplication Compatibility

If \(A\) is \(3\times4\) and \(B\) is \(4\times2\), is \(AB\) defined? If so, what are its dimensions?

Check Answer

Yes. The inner dimensions match:

\[ 4=4. \]

The result has dimensions

\[ \boxed{3\times2}. \]

5. Undefined Product

If \(A\) is \(2\times3\) and \(B\) is \(4\times2\), is \(AB\) defined?

Check Answer

No.

The inner dimensions are

\[ 3 \]

and

\[ 4, \]

which do not match.

6. Two-by-Two Product

Compute

\[ \begin{bmatrix} 1 & 2\\ 0 & 1 \end{bmatrix} \begin{bmatrix} 3 & 4\\ 5 & 6 \end{bmatrix}. \]
Check Answer \[ \boxed{ \begin{bmatrix} 13 & 16\\ 5 & 6 \end{bmatrix} }. \]

7. Matrix–Vector Product

Compute

\[ \begin{bmatrix} 2 & 3\\ 4 & -1 \end{bmatrix} \begin{bmatrix} 1\\ 2 \end{bmatrix}. \]
Check Answer \[ \begin{bmatrix} 2(1)+3(2)\\ 4(1)-1(2) \end{bmatrix} = \boxed{ \begin{bmatrix} 8\\ 2 \end{bmatrix} }. \]

8. Noncommutativity

Is it always true that

\[ AB=BA? \]
Check Answer \[ \boxed{\text{No.}} \]

Matrix multiplication is generally not commutative.

9. Identity Matrix

What is

\[ I_2 \begin{bmatrix} 4 & 1\\ 2 & 3 \end{bmatrix}? \]
Check Answer \[ \boxed{ \begin{bmatrix} 4 & 1\\ 2 & 3 \end{bmatrix} }. \]

10. Matrix Square

Compute

\[ A^2 \]

for

\[ A= \begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix}. \]
Check Answer \[ A^2= AA= \boxed{ \begin{bmatrix} 1 & 2\\ 0 & 1 \end{bmatrix} }. \]

11. Data Matrix Product

Suppose

\[ Q= \begin{bmatrix} 2 & 3\\ 4 & 1 \end{bmatrix} \]

and

\[ \mathbf{p}= \begin{bmatrix} 10\\ 5 \end{bmatrix}. \]

Compute \(Q\mathbf{p}\).

Check Answer \[ Q\mathbf{p} = \begin{bmatrix} 2(10)+3(5)\\ 4(10)+1(5) \end{bmatrix} = \boxed{ \begin{bmatrix} 35\\ 45 \end{bmatrix} }. \]

12. Product Dimensions

If

\[ A \]

is \(5\times2\) and

\[ B \]

is \(2\times7\), what are the dimensions of \(AB\)?

Check Answer \[ \boxed{5\times7}. \]

Key Ideas

Where This Leads

Matrix arithmetic turns matrices from passive containers into active algebraic objects. Addition combines compatible structures, scalar multiplication rescales them, and matrix multiplication composes linear relationships in a way that preserves row-column meaning.

From here, the section can develop matrix equations, identity and inverse matrices, determinants, invertibility, powers, transformations, and rank. Matrix multiplication will also become essential in geometry, recurrence systems, computer graphics, statistics, numerical methods, and later linear algebra.