Subsemigroup/Examples/2x2 Matrices with One Non-Zero Entry

Example of Subsemigroup

Let $\struct {S, \times}$ be the semigroup formed by the set of order $2$ square matrices over the real numbers $R$ under (conventional) matrix multiplication.

Let $T$ be the subset of $S$ consisting of the matrices of the form $\begin{bmatrix} x & 0 \\ 0 & 0 \end{bmatrix}$ for $x \in \R$.

Then $\struct {T, \times}$ is a subsemigroup of $\struct {S, \times}$.

Proof

From the Subsemigroup Closure Test it is sufficient to demonstrate that $\struct {T, \times}$ is closed.

Let $A = \begin{bmatrix} x & 0 \\ 0 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} y & 0 \\ 0 & 0 \end{bmatrix}$.

Then:

\(\ds A B\)

\(=\)

\(\ds \begin{bmatrix} x & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} y & 0 \\ 0 & 0 \end{bmatrix}\)

\(\ds \)

\(=\)

\(\ds \begin{bmatrix} x y + 0 \times 0 & x \times 0 + 0 \times 0 \\ 0 \times y + 0 \times 0 & 0 \times 0 + 0 \times 0 \end{bmatrix}\)

Definition of Matrix Product (Conventional)

\(\ds \)

\(=\)

\(\ds \begin{bmatrix} x y & 0 \\ 0 & 0 \end{bmatrix}\)

\(\ds \)

\(=\)

\(\ds T\)

Hence the result.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 32$ Identity element and inverses