已知 $\displaystyle\begin{pmatrix}9 & 2\\[4pt]3 & 0\end{pmatrix}\begin{pmatrix}1\\[4pt]-2y\end{pmatrix}=\begin{pmatrix}x\\[4pt]2x-7\end{pmatrix}$，求 $x$ 和 $y$ 的值。 罗伯特（Robert）喜欢吃牛肉汉堡和鸡肉汉堡。
矩阵 $T$ 表示他在特定一周内食用的牛肉汉堡和鸡肉汉堡的总数量。 牛肉     鸡肉 $T=\displaystyle\begin{pmatrix}5 & 3\\[4pt]1 & 2\end{pmatrix}$ 周一至周五 周六和周日 一个牛肉汉堡含有 300 卡路里，一个鸡肉汉堡含有 270 卡路里。 用一个 2 x 1 的列矩阵 C 表示这些卡路里。 计算矩阵 $N = TC$。 说明矩阵 $N$ 中的元素代表什么。 计算矩阵 $A=\frac{1}{7}\begin{pmatrix}1 & 1\end{pmatrix}N$。 答案 $A=\begin{pmatrix} \;\;\;\;\end{pmatrix}$ 解释矩阵 $A$ 代表什么。 答案 ... ... 罗伯特希望在工作日将他的总卡路里摄入量减少 50%，并在周末保持他的总卡路里摄入量。 使用矩阵乘法，计算他在接下来的一周内应该摄入的卡路里总量。 答案 ...

1. Multiply the matrix by the column vector.

$\begin{pmatrix}9&2\\3&0\end{pmatrix}\begin{pmatrix}1\\-2y\end{pmatrix}=\begin{pmatrix}9(1)+2(-2y)\\3(1)+0(-2y)\end{pmatrix}=\begin{pmatrix}9-4y\\3\end{pmatrix}$

2. Equate corresponding entries and solve for $x$ and $y$.

Since
$\begin{pmatrix}9-4y\\3\end{pmatrix}=\begin{pmatrix}x\\2x-7\end{pmatrix},$
compare the second entries:
$3=2x-7$
$2x=10$
$x=5$
Compare the first entries:
$9-4y=x$
$9-4y=5$
$-4y=-4$
$y=1$

1. Form the $2\times 1$ column matrix using the given calorie values in Beef–Chicken order.

Since the columns of $T$ are ordered as Beef, then Chicken,
$C=\begin{pmatrix}\text{calories in a Beef-burger}\\\text{calories in a Chicken-burger}\end{pmatrix}=\begin{pmatrix}300\\270\end{pmatrix}.$

1. Substitute the matrices $T$ and $C$ into $N=TC$.

From the question and part (b)(i),
$T=\begin{pmatrix}5&3\\1&2\end{pmatrix},\quad C=\begin{pmatrix}300\\270\end{pmatrix}$
So
$N=\begin{pmatrix}5&3\\1&2\end{pmatrix}\begin{pmatrix}300\\270\end{pmatrix}.$

2. Carry out row-by-column multiplication to find each entry of $N$.

$N=\begin{pmatrix}5(300)+3(270)\\1(300)+2(270)\end{pmatrix}$
$=\begin{pmatrix}1500+810\\300+540\end{pmatrix}$
$=\begin{pmatrix}2310\\840\end{pmatrix}.$

1. Interpret each entry of $N$ using the row categories of $T$.

Since
$N=TC=\begin{pmatrix}2310\\840\end{pmatrix},$
the first entry corresponds to the first row of $T$, which is Monday to Friday, and the second entry corresponds to the second row of $T$, which is Saturday and Sunday.

1. Substitute $N$ and evaluate the matrix product and scalar multiplication.

From part (b)(ii)(a),
$N=\begin{pmatrix}2310\\840\end{pmatrix}.$
$A=\frac{1}{7}\begin{pmatrix}1&1\end{pmatrix}\begin{pmatrix}2310\\840\end{pmatrix}$
$=\frac{1}{7}\begin{pmatrix}1(2310)+1(840)\end{pmatrix}$
$=\frac{1}{7}\begin{pmatrix}3150\end{pmatrix}$
$=\begin{pmatrix}450\end{pmatrix}$

1. Interpret the matrix operation used to obtain $A$.

Since $A=\frac{1}{7}\begin{pmatrix}1&1\end{pmatrix}N,$ the entries of $N$ are added to give the total calories for the whole week, then divided by $7$. Hence $A=\begin{pmatrix}450\end{pmatrix}$ represents the average number of calories consumed per day from the burgers in that week.

1. Form the matrix multiplication representing 50% of weekday calories and 100% of weekend calories.

From earlier, $N=\begin{pmatrix}2310\\840\end{pmatrix}.$ Since weekday intake is reduced by $50\%$, Robert consumes $50\%=0.5$ of the weekday calories. Weekend intake is maintained, so he consumes $100\%=1$ of the weekend calories.

$\text{Total}=\begin{pmatrix}0.5&1\end{pmatrix}\begin{pmatrix}2310\\840\end{pmatrix}$

2. Evaluate the matrix product to obtain the total calorie intake.

$\begin{pmatrix}0.5&1\end{pmatrix}\begin{pmatrix}2310\\840\end{pmatrix}=0.5(2310)+1(840)$
$=1155+840=1995$