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.