Example · Undergraduate
Terms
| Term | This example | Why |
|---|
| associative |
✓ |
Multiplication of $2 \times 2$ matrices is associative, because $(A B) C$ and $A (B C)$ both represent the composition of the same three linear maps.
For example, multiplying the first two matrices first and multiplying the last two matrices first give the same product:
\[\left(
\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}
\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}
\right)
\begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}
= \begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}
\begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}
= \begin{bmatrix} 4 & 1 \\ 2 & 1 \end{bmatrix},\]
\[\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}
\left(
\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}
\begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}
\right)
= \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}
\begin{bmatrix} 2 & 0 \\ 2 & 1 \end{bmatrix}
= \begin{bmatrix} 4 & 1 \\ 2 & 1 \end{bmatrix}.\]
|
| commutative |
✗ |
Multiplication of $2 \times 2$ matrices is not commutative, because multiplying the same two matrices in opposite orders can give different products:
\[\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}
\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}
= \begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix},
\qquad
\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}
\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}
= \begin{bmatrix} 1 & 1 \\ 1 & 2 \end{bmatrix}.\]
|
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