3.1 Matrix multiplication and identity matrices

Matrix multiplication is the first matrix operation that genuinely mixes rows with columns. It is also the operation that lets matrices encode composition, systems of equations, and later inverse matrices. Because of that, you should not memorize the rule as a pattern of symbols only. You should know what the dimensions are doing at each step.

Why multiplication is more subtle than addition

Addition and scalar multiplication act entry by entry. Matrix multiplication is different. To compute one output entry, you compare one row of the left matrix with one column of the right matrix.

That is why dimensions matter so strictly.

The inner dimensions must match. The outer dimensions tell you the size of the result.

The row-by-column rule

This rule explains three important facts at once:

• multiplication is not entrywise;
• the inner dimensions must match;
• the output entry uses every matched position in the row and column.

Matrix-vector multiplication is a system statement

If x is a column vector, then Ax is a special case of matrix multiplication. It packages the left-hand sides of a linear system into one object.

For

$$A = \begin{bmatrix} 1 & 2 & -1 \\ 3 & -1 & 5 \end{bmatrix}, \qquad x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix},$$

we have

$$Ax = \begin{bmatrix} x_1 + 2x_2 - x_3 \\ 3x_1 - x_2 + 5x_3 \end{bmatrix}.$$

So the system $Ax = b$ is not merely shorthand. It is a matrix product whose entries reproduce the equations of the system.

Identity matrices do nothing, on purpose

For example,

$$I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \qquad I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}.$$

The identity matrix matters because it preserves any compatible matrix:

$$AI_n = A, \qquad I_m A = A$$

whenever the sizes match.

That is exactly why inverse matrices are defined through the identity later: if $A^{-1}$ exists, then $AA^{-1} = I$.

Multiplication is usually not commutative

One of the first conceptual shocks in linear algebra is that

$$AB \ne BA$$

in general.

Sometimes both products are defined and differ. Sometimes one product is defined and the other is not. So order matters twice: it matters for meaning, and it matters for the final answer.

Use the figure below to watch one output entry being built from a selected row and a selected column.

Read the product by columns as well as by entries

The row-by-column rule is the standard local computation rule, but it is not the only useful interpretation.

Write the columns of $B$ as

$$B = [b_1\ b_2\ \cdots\ b_p].$$

Then the product can be read as

$$AB = [Ab_1\ Ab_2\ \cdots\ Ab_p].$$

So each column of AB is obtained by applying $A$ to the corresponding column of $B$.

Matrix multiplication represents composition

The multiplication rule is not arbitrary. It is the rule that makes matrices encode linear transformations in sequence.

If a vector x is first sent to Bx, and then that result is sent to A(Bx), the combined effect is

$$(AB)x.$$

That is why the inner dimensions must match. The output of the first map must be a valid input for the second one.

This does not mean that order may be changed. Associativity lets us change parentheses, not the order of the factors themselves.

Standard basis vectors explain why columns behave so cleanly

The standard basis vectors make the column interpretation precise. In $R^n$, the vector $e_k$ has a 1 in position k and 0 everywhere else. If $A$ is an $m \times n$ matrix, then $Ae_k$ is exactly the kth column of $A$.

This is why the identity matrix behaves so naturally. The columns of $I_n$ are $e_1, e_2, \ldots, e_n$, so right-multiplying by $I_n$ simply reproduces the columns of $A$ one by one.

This also explains why a compatible zero matrix on the right forces the product to be zero: every column of the zero matrix is the zero vector, so every column of the product is $A0 = 0$.

The first algebra laws worth remembering

Once multiplication is defined, the next issue is how it interacts with the other matrix operations you already know.

Whenever the sizes are compatible, matrix multiplication satisfies:

$$A(B + C) = AB + AC, \qquad (A + B)C = AC + BC,$$

and scalar multiplication may be moved in or out:

$$(cA)B = c(AB) = A(cB).$$

The zero matrix is the simplest sanity check for these rules. If 0 is a compatible zero matrix, then

$$A0 = 0, \qquad 0A = 0.$$

The reason is that every row-by-column product uses only zero entries from the zero matrix, so every output entry is zero as well.

These identities are basic, but they matter because later arguments about inverse matrices, row operations, and block-matrix computation assume them silently. If you do not know them explicitly, longer calculations become much harder to audit.

Common mistakes

Quick checks

Exercise

Related notes

This note depends on 2.1 Matrix basics. Continue to 3.2 Transpose and special matrices or jump ahead to 5.1 Invertible matrices.