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.

Answer 1

\[LetA = \begin{bmatrix}a_1 & 0 \\ 0 & b_1\end{bmatrix}, B = \begin{bmatrix}a_2 & 0 \\ 0 & b_2\end{bmatrix} \in M\] 
\[A * B = AB\] 
\[ = \begin{bmatrix}a_1 & 0 \\ 0 & b_1\end{bmatrix}\begin{bmatrix}a_2 & 0 \\ 0 & b_2\end{bmatrix}\] 
\[ = \begin{bmatrix}a_1 a_2 & 0 \\ 0 & b_1 b_2\end{bmatrix} \in M, \left( \because a_1 a_2 \text{ and } b_1 b_2 \in R - \left\{ 0 \right\} \right)\] 
\[\text{Therefore},\] 
\[A * B \in M, \forall A, B \in M\]

Thus, * is a binary operation on M.