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Question
Prove that the operation * on the set
\[M = \left\{ \begin{bmatrix}a & 0 \\ 0 & b\end{bmatrix}; a, b \in R - \left\{ 0 \right\} \right\}\] defined by A * B = AB is a binary operation.
Solution
\[LetA = \begin{bmatrix}a_1 & 0 \\ 0 & b_1\end{bmatrix}, B = \begin{bmatrix}a_2 & 0 \\ 0 & b_2\end{bmatrix} \in M\]
\[A * B = AB\]
\[ = \begin{bmatrix}a_1 & 0 \\ 0 & b_1\end{bmatrix}\begin{bmatrix}a_2 & 0 \\ 0 & b_2\end{bmatrix}\]
\[ = \begin{bmatrix}a_1 a_2 & 0 \\ 0 & b_1 b_2\end{bmatrix} \in M, \left( \because a_1 a_2 \text{ and } b_1 b_2 \in R - \left\{ 0 \right\} \right)\]
\[\text{Therefore},\]
\[A * B \in M, \forall A, B \in M\]
Thus, * is a binary operation on M.
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