Answer
a. See explanations.
b. $\begin{bmatrix} 84 & 87.2 \\ 79 & 81 \\ 90 & 88.4 \\ 73 & 68.6 \\69 & 73.4 \end{bmatrix}$
System-1: Student-1: B; Student-2: C; Student-3: A; Student-4: C; Student-5: D
System-2: Student-1: B; Student-2: B; Student-3: B; Student-4: D; Student-5: C
Work Step by Step
a. The grading system that is represented by matrix B are weight factors: system-1 indicates that the Midterm and Final contribute $50\%$ each, while system-2 indicates that Midterm counts for $30\%$ and the Final counts for $70\%$.
b. Using the given matrices, we have
$AB=\begin{bmatrix} 76 & 92 \\ 74 & 84 \\ 94 & 86 \\ 84 & 62 \\58 & 80 \end{bmatrix}\begin{bmatrix} 0.5 & 0.3 \\ 0.5 & 0.7 \end{bmatrix}=\begin{bmatrix} 76(0.5)+92(0.5) & 76(0.3)+92(0.7) \\ 74(0.5)+84(0.5) & 74(0.3)+84(0.7) \\ 94(0.5)+86(0.5) & 94(0.3)+86(0.7) \\ 84(0.5)+62(0.5) & 84(0.3)+62(0.7) \\58(0.5)+80(0.5) & 58(0.3)+80(0.7) \end{bmatrix}=\begin{bmatrix} 84 & 87.2 \\ 79 & 81 \\ 90 & 88.4 \\ 73 & 68.6 \\69 & 73.4 \end{bmatrix}$
We have the following student scores with the given systems:
System-1 (first column):
Student-1: B; Student-2: C; Student-3: A; Student-4: C; Student-5: D
System-2 (second column):
Student-1: B; Student-2: B; Student-3: B; Student-4: D; Student-5: C
