Chapter 5 - Number Theory and the Real Number System - 5.5 Real Numbers and Their Properties; Clock Addition - Exercise Set 5.5 - Page 310: 66

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(a) Find \[\left[ \begin{matrix} 2 & 3 \\ 4 & 7 \\ \end{matrix} \right]\times \left[ \begin{matrix} 0 & 1 \\ 5 & 6 \\ \end{matrix} \right]\]as, \[\begin{align} & \left[ \begin{matrix} 2 & 3 \\ 4 & 7 \\ \end{matrix} \right]\times \left[ \begin{matrix} 0 & 1 \\ 5 & 6 \\ \end{matrix} \right]=\left[ \begin{matrix} 2\cdot 0+3\cdot 5 & 2\cdot 1+3\cdot 6 \\ 4\cdot 0+7\cdot 5 & 4\cdot 1+7\cdot 6 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 0+15 & 2+18 \\ 0+35 & 4+42 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 15 & 20 \\ 35 & 46 \\ \end{matrix} \right] \end{align}\] Hence, \[\left[ \begin{matrix} 2 & 3 \\ 4 & 7 \\ \end{matrix} \right]\times \left[ \begin{matrix} 0 & 1 \\ 5 & 6 \\ \end{matrix} \right]=\left[ \begin{matrix} 15 & 20 \\ 35 & 46 \\ \end{matrix} \right]\]. (b) Find \[\left[ \begin{matrix} 0 & 1 \\ 5 & 6 \\ \end{matrix} \right]\times \left[ \begin{matrix} 2 & 3 \\ 4 & 7 \\ \end{matrix} \right]\]as, \[\begin{align} & \left[ \begin{matrix} 0 & 1 \\ 5 & 6 \\ \end{matrix} \right]\times \left[ \begin{matrix} 2 & 3 \\ 4 & 7 \\ \end{matrix} \right]=\left[ \begin{matrix} 0\cdot 2+1\cdot 4 & 0\cdot 3+1\cdot 7 \\ 5\cdot 2+6\cdot 4 & 5\cdot 3+6\cdot 7 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 0+4 & 0+7 \\ 10+24 & 15+42 \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} 4 & 7 \\ 34 & 57 \\ \end{matrix} \right] \end{align}\] Hence, \[\left[ \begin{matrix} 0 & 1 \\ 5 & 6 \\ \end{matrix} \right]\times \left[ \begin{matrix} 2 & 3 \\ 4 & 7 \\ \end{matrix} \right]=\left[ \begin{matrix} 4 & 7 \\ 34 & 57 \\ \end{matrix} \right]\]. (c) The row of first set is multiplied with the column of second set. Thus, the row of first is distributed over column of second. Hence, distributive property is used in above multiplication.