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Invertible Element Binary Operation

Prove that th...

Question

Prove that the operation * on the set

$M = \{[\begin{matrix} a & 0 \\ 0 & b \end{matrix}]; a, b \in R - \{0\}\}$ defined by A * B = AB is a binary operation.

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Solution

$Let A = [\begin{matrix} a_{1} & 0 \\ 0 & b_{1} \end{matrix}], B = [\begin{matrix} a_{2} & 0 \\ 0 & b_{2} \end{matrix}] \in M A * B = A B = [\begin{matrix} a_{1} & 0 \\ 0 & b_{1} \end{matrix}] [\begin{matrix} a_{2} & 0 \\ 0 & b_{2} \end{matrix}] = [\begin{matrix} a_{1} a_{2} & 0 \\ 0 & b_{1} b_{2} \end{matrix}] \in M, (∵ a_{1} a_{2} and b_{1} b_{2} \in R - \{0\}) Therefore, A * B \in M, \forall A, B \in M$

Thus, * is a binary operation on M.

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