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Prove that the operation is a binary operation
Given:
Set \( M = \left\{ \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} : a, b \in \mathbb{R} – \{0\} \right\} \)
Operation defined by:
\( A * B = AB \)
Concept:
To prove a binary operation, we must show closure, i.e., \( A, B \in M \Rightarrow AB \in M \)
Solution:
Let:
\[
A =
\begin{bmatrix}
a_1 & 0 \\
0 & b_1
\end{bmatrix},
\quad
B =
\begin{bmatrix}
a_2 & 0 \\
0 & b_2
\end{bmatrix}
\]
Then:
\[
AB =
\begin{bmatrix}
a_1 a_2 & 0 \\
0 & b_1 b_2
\end{bmatrix}
\]
Since:
- \( a_1 a_2 \neq 0 \)
- \( b_1 b_2 \neq 0 \)
we have:
\[
AB \in M
\]
Conclusion:
The set is closed under the operation.
✔ Therefore, \( A * B = AB \) defines a binary operation on \( M \).