The tables below show the times by three runners in two heats of a race. Write a matrix that shows the increase (decrease) in each time from the first heat to the second. \begin{tabular}{r|rrr|r} Heat 1 & \begin{tabular}{r} Time \\ $(\mathrm{sec})$ \end{tabular} & & Heat 2 & \multicolumn{1}{c}{\begin{tabular}{c} Time \\ $(\mathrm{sec})$ \end{tabular}} \\ \hline Russell & 14.5 & & Russell & 14.3 \\ Sergy & 14.8 & Sergy & 14.4 \\ Omar & 15.2 & & Omar & 14.7 \end{tabular}
Solution
Step 1 ：The tables below show the times by three runners in two heats of a race. We are asked to write a matrix that shows the increase (decrease) in each time from the first heat to the second.
Step 2 ：This can be calculated by subtracting the time of the first heat from the time of the second heat for each runner.
Step 3 ：The times for the first heat are: Russell: 14.5 sec, Sergy: 14.8 sec, Omar: 15.2 sec.
Step 4 ：The times for the second heat are: Russell: 14.3 sec, Sergy: 14.4 sec, Omar: 14.7 sec.
Step 5 ：By subtracting the times of the first heat from the times of the second heat, we get the following differences: Russell: -0.2 sec, Sergy: -0.4 sec, Omar: -0.5 sec.
Step 6 ：These differences can be represented as a matrix: \(\begin{bmatrix} -0.2 \ -0.4 \ -0.5 \end{bmatrix}\).
Step 7 ：The negative values indicate a decrease in time, which means the runner was faster in the second heat.
Step 8 ：\(\boxed{\text{Final Answer: The matrix that shows the increase (decrease) in each time from the first heat to the second is } \begin{bmatrix} -0.2 \ -0.4 \ -0.5 \end{bmatrix}}\)
