Matrix Multiplication

This lesson covers matrix multiplication — one of the most important operations in linear algebra. Unlike addition, matrix multiplication is not element-by-element; it follows a specific "row times column" rule and has some surprising properties.

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The Rule for Matrix Multiplication

To multiply matrices $A$A (order $m \times n$m×n) and $B$B (order $n \times p$n×p):

• The number of columns of A must equal the number of rows of B
• The result $AB$AB has order $m \times p$m×p
• Entry $(i, j)$(i,j) of $AB$AB = dot product of row $i$i of $A$A with column $j$j of $B$B

$$(AB)_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}$$(AB)ij​=k=1∑n​aik​bkj​

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2×2 Matrix Multiplication

$$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} ae+bg & af+bh \\ ce+dg & cf+dh \end{pmatrix}$$(ac​bd​)(eg​fh​)=(ae+bgce+dg​af+bhcf+dh​)

Worked Example 1:

$$\begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \begin{pmatrix} 5 & -1 \\ 0 & 2 \end{pmatrix} = \begin{pmatrix} 10+0 & -2+6 \\ 5+0 & -1+8 \end{pmatrix} = \begin{pmatrix} 10 & 4 \\ 5 & 7 \end{pmatrix}$$(21​34​)(50​−12​)=(10+05+0​−2+6−1+8​)=(105​47​)

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Matrix Multiplication Is NOT Commutative

In general, $AB \neq BA$AB=BA.

Example: Let $A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$A=(10​21​) and $B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$B=(01​10​).

$AB = \begin{pmatrix} 2 & 1 \\ 1 & 0 \end{pmatrix}$AB=(21​10​), but $BA = \begin{pmatrix} 0 & 1 \\ 1 & 2 \end{pmatrix}$BA=(01​12​).

$AB \neq BA$AB=BA — matrix multiplication is not commutative.

Exam Tip: Never assume you can swap the order of matrix multiplication. This is one of the most common errors.

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Properties of Matrix Multiplication

Property	Statement
Associative	$(AB)C = A(BC)$(AB)C=A(BC)
Distributive (left)	$A(B + C) = AB + AC$A(B+C)=AB+AC
Distributive (right)	$(A + B)C = AC + BC$(A+B)C=AC+BC
Identity	$AI = IA = A$AI=IA=A
Scalar	$k(AB) = (kA)B = A(kB)$k(AB)=(kA)B=A(kB)
NOT commutative	$AB \neq BA$AB=BA in general
NOT cancellative	$AB = AC$AB=AC does not imply $B = C$B=C in general

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Multiplying with the Identity Matrix

The identity matrix $I$I acts like the number 1 in multiplication:

$$AI = IA = A$$AI=IA=A

For $2 \times 2$2×2: $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$I=(10​01​). For $3 \times 3$3×3: $I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$I=​100​010​001​​.

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Matrix Powers

For a square matrix $A$A:

$A^0 = I, \quad A^1 = A, \quad A^2 = AA, \quad A^3 = A^2 \cdot A, \quad \ldots$A0=I,A1=A,A2=AA,A3=A2⋅A,…