1.7矩阵

考虑以下矩阵:

A=2-14-1           B=11-4k{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="黑体斜体A =开括号表第2行单元格负1端单元格第4行单元格负1端单元格结束表闭括号空白黑体斜体B =开括号表第1行单元格负4端单元格k结束表闭括号" style="vertical-align:-21px;height:54px;width:231px" loading="lazy">

在哪里k∈ℝ{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="K个实数元素" style="vertical-align:-6px;height:22px;width:40px" loading="lazy">是常数。

让I{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="加粗斜体I" style="vertical-align:-6px;height:22px;width:9px" loading="lazy">是2×2{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="2叉乘2" style="vertical-align:-6px;height:22px;width:36px" loading="lazy">单位矩阵，让0是2×2 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="2叉乘2空间" style="vertical-align:-6px;height:22px;width:40px" loading="lazy">零矩阵。

(我)B2{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体B的2次方" style="vertical-align:-6px;height:23px;width:22px" loading="lazy"> (2)A-2I{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="加粗的斜体A减去加粗的2加粗的斜体I" style="vertical-align:-6px;height:22px;width:52px" loading="lazy"> (3)AB{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="加粗的斜体" style="vertical-align:-6px;height:22px;width:29px" loading="lazy">(iv)BA{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体的;粗体的" style="vertical-align:-6px;height:22px;width:29px" loading="lazy">(v)A-1B{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="加粗的斜体A的加粗- 1的幂最后的指数加粗的斜体B" style="vertical-align:-6px;height:23px;width:48px" loading="lazy">

C{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体C" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">而且D{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体D" style="vertical-align:-6px;height:22px;width:16px" loading="lazy">是这样的矩阵A+C=0{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="加粗斜体A加上加粗斜体C等于加粗0" style="vertical-align:-6px;height:22px;width:75px" loading="lazy">而且B-1+D= 0{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体B的- 1末端指数次方加上粗体D等于粗体 粗体0" style="vertical-align:-6px;height:23px;width:97px" loading="lazy">

(i)写下矩阵C{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体C" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">．

(2)

找到矩阵D{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体D" style="vertical-align:-6px;height:22px;width:16px" loading="lazy">．

考虑矩阵

A=3-8p7{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体A等于开括号表第3行单元格- 8结束单元格第p行结束表第7结束括号" style="vertical-align:-21px;height:54px;width:95px" loading="lazy">

在哪里 p∈ℝ{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="空格p是直线实数的元素" style="vertical-align:-6px;height:22px;width:44px" loading="lazy">是常数。

考虑到det A=-3{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="det空格加粗斜体A = - 3" style="vertical-align:-6px;height:22px;width:85px" loading="lazy">，求的值 p{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="空间p" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">．

考虑这两个矩阵

B=4q-13       and       C=r113{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体B等于开括号表第4行q行cell负1端cell 3结束表右括号空白和空白粗体斜体C等于开括号表第r行1行3结束表右括号" style="vertical-align:-21px;height:54px;width:271px" loading="lazy">

在哪里q, r∈ℝ{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="Q，空白r元素的实数" style="vertical-align:-6px;height:22px;width:56px" loading="lazy">是常数。

考虑到BC=CB{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="加粗斜体B加粗斜体C等于加粗斜体C加粗斜体B" style="vertical-align:-6px;height:22px;width:72px" loading="lazy">的值q{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="问" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">而且r{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="r" style="vertical-align:-6px;height:22px;width:8px" loading="lazy">．

考虑矩阵 P=2-131{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="加粗空格加粗斜体P等于开括号表第2行单元格- 1结束单元格第3行1结束表闭括号" style="vertical-align:-21px;height:54px;width:96px" loading="lazy">而且D=1000.8{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体D等于开括号表第1行0行0单元格0.8结束单元格结束表闭括号" style="vertical-align:-21px;height:54px;width:93px" loading="lazy">．

(i)的行列式P{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="加粗斜体P" style="vertical-align:-6px;height:22px;width:13px" loading="lazy">

(2)P-1{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗斜体P的- 1次幂的末端指数" style="vertical-align:-6px;height:23px;width:30px" loading="lazy">．

考虑矩阵乘积PDP-1{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="加粗的斜体P加粗的斜体D加粗的斜体P的- 1次幂末端指数" style="vertical-align:-6px;height:23px;width:55px" loading="lazy">．

(我)

表明,PDP-12=PD2P-1{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="开括号加粗斜体P加粗斜体D加粗斜体P的负1次末端指数次方闭括号平方等于加粗斜体P加粗斜体D的平方加粗斜体P的负1次末端指数次方" style="vertical-align:-6px;height:23px;width:153px" loading="lazy">而且PDP-13=PD3P-1{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="开括号加粗斜体P加粗斜体D加粗斜体P的负1次末端指数次方闭括号的立方等于加粗斜体P加粗斜体D的立方加粗斜体P的负1次末端指数次方" style="vertical-align:-6px;height:23px;width:153px" loading="lazy">．

(2)

使用(b)(i)部分的结果提出的表达式PDP-1n{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="左括号加粗斜体P加粗斜体D加粗斜体P的- 1次幂结束指数闭括号的n次幂" style="vertical-align:-6px;height:23px;width:75px" loading="lazy">在这方面P{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="加粗斜体P" style="vertical-align:-6px;height:22px;width:13px" loading="lazy"> ，D{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体D" style="vertical-align:-6px;height:22px;width:16px" loading="lazy">而且P-1{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗斜体P的- 1次幂的末端指数" style="vertical-align:-6px;height:23px;width:30px" loading="lazy">．

动态系统的转换矩阵为T=0.880.080.120.92{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体T =开括号表行cell 0.88 end cell cell 0.08 end cell行cell 0.12 end cell cell 0.92 end cell结束表闭括号" style="vertical-align:-21px;height:54px;width:121px" loading="lazy">．

找到T5{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体T的5次方" style="vertical-align:-6px;height:23px;width:21px" loading="lazy">．

有一家人晚餐要买汉堡，想点三个素汉堡、一个牛肉汉堡和两个鸡肉汉堡。

三个素食汉堡总共要15.45美元，一个牛肉汉堡要6.15美元，两个鸡肉汉堡总共要11.90美元。

(我)

行矩阵，Q{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体Q" style="vertical-align:-6px;height:22px;width:16px" loading="lazy">，表示该家庭希望购买的每种汉堡的数量

(2)

一个列矩阵，PA{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体P下标A" style="vertical-align:-12px;height:28px;width:23px" loading="lazy">，表示A店每种汉堡的价格。

B汉堡店的素食汉堡、牛肉汉堡和鸡肉汉堡的售价分别为4.75美元、5.85美元和5.50美元。

(我)

写下一个列矩阵，PB{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体P下标B" style="vertical-align:-12px;height:28px;width:23px" loading="lazy">，表示B店每种汉堡的价格。

(2)

因此，写下一个成本矩阵，C{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体C" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">，表示两家门店的成本。

café网站正在招聘一名值班经理、两名咖啡师、一名洗碗工和三名服务员。他们决定在社交媒体上发布招聘广告，也可以通过招聘机构。他们收到了123份来自社交媒体上招聘广告的申请，57份来自招聘机构的申请。

写下一个列矩阵，C{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体C" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">，代表他们通过在社交媒体上发布招聘广告和通过招聘机构收到的申请数量。

总的来说，22%的求职者申请了值班经理的工作，28%申请了咖啡师的工作，18%申请了洗碗工的工作，剩下的人申请了服务员的工作。请注意，每个申请人只能申请一个可用的工作。

写下一个行矩阵，R{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="加粗斜体R" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">，表示申请每个不同职位的申请人的百分比。

一旦café为每个职位选择了合适的候选人，每个新员工每周总共工作40个小时。值班经理和咖啡师的时薪是每小时20美元，洗碗工和服务员的时薪是17.25美元。

阿曼达是一个房东，她有新房客搬进了三个没有家具的新房子。阿曼达在当地一家小家具店得到了一笔特别的交易，所以她向她的租户提出，她将代表他们购买一些桌子、椅子和/或沙发，前提是他们会偿还她的费用。下表显示了每个房子订购的不同项目的数量。

(我)

写下3×3{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="3叉乘3" style="vertical-align:-6px;height:22px;width:36px" loading="lazy">矩阵,F{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体F" style="vertical-align:-6px;height:22px;width:13px" loading="lazy">，表示这三家的家具订单。

(2)

写下1×3{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="1乘以3" style="vertical-align:-6px;height:22px;width:36px" loading="lazy">行矩阵,S{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体S" style="vertical-align:-6px;height:22px;width:13px" loading="lazy">，代表阿曼达在家具店的总购物清单。

阿曼达为每件物品支付的价格见下表。

(我)

通过对矩阵进行适当的乘法运算F{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体F" style="vertical-align:-6px;height:22px;width:13px" loading="lazy">，找出每家欠阿曼达的总金额。

(2)

通过对矩阵进行适当的乘法运算S{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="粗体斜体S" style="vertical-align:-6px;height:22px;width:13px" loading="lazy"> ，找出阿曼达付给家具店的总金额。

一个人在城市中支付的公交费用取决于这个人是学生、成年人还是养老金领取者。

下表显示了三辆不同巴士在某一天的总票价，以及每种票价的支付金额。

让s, a{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="S，空格a" style="vertical-align:-6px;height:22px;width:27px" loading="lazy">而且 p{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="空间p" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">代表学生、成年人和领取养老金者分别支付的金额。

写出一个由三个线性方程组组成的方程组s, a{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="S，空格a" style="vertical-align:-6px;height:22px;width:27px" loading="lazy">而且 p{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="空间p" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">表示上表中显示的信息。

当天，D路共售出112张学生票、91张成人票和22张养老金票。

玩家在3周内登录《使命召唤》、《FIFA》或《刺客信条》的次数如下表所示。

玩家每周花在游戏上的总时间如下表所示。

该玩家从未同时登录多款游戏。

玩家认为，对于每款游戏，在这三周内每次登录会话所花的平均时间是一致的。

二次函数的图 f(x)=ax2+bx+c{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="空格f左括号x右括号等于ax方加bx加c" style="vertical-align:-6px;height:23px;width:140px" loading="lazy">通过这些点(-1,11), (3,-5){"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="左括号- 1,11右括号，空格左括号3，- 5右括号" style="vertical-align:-6px;height:22px;width:123px" loading="lazy">而且(6, 4){"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="左括号6空格4右括号" style="vertical-align:-6px;height:22px;width:40px" loading="lazy">．

(a)证明这一点a{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="一个" style="vertical-align:-6px;height:22px;width:10px" loading="lazy">，b{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="b" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">而且c{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="c" style="vertical-align:-6px;height:22px;width:10px" loading="lazy">必须满足下列线性方程组:

a-b+c=11{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="A - b + c = 11" style="vertical-align:-6px;height:22px;width:98px" loading="lazy">

9a+3b+c=-5{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="9a + 3b + c = - 5" style="vertical-align:-6px;height:22px;width:124px" loading="lazy">

36a+6b+c=4{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="36a + 6b + c = 4" style="vertical-align:-6px;height:22px;width:116px" loading="lazy">

函数的图像 f(x)=ax2+bx+c{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="空格f左括号x右括号等于ax方加bx加c" style="vertical-align:-6px;height:23px;width:140px" loading="lazy">通过这些点(-1, 5), (3,1) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="左括号- 1，空格5右括号，空格，左括号，3,1右括号空格" style="vertical-align:-6px;height:22px;width:105px" loading="lazy">而且(4,-5){"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="左括号4，右括号- 5" style="vertical-align:-6px;height:22px;width:53px" loading="lazy">．

写出一个线性方程组a, b{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="A，空格b" style="vertical-align:-6px;height:22px;width:29px" loading="lazy">而且c{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="c" style="vertical-align:-6px;height:22px;width:10px" loading="lazy">必须满足。

下表显示了三个不同农场在给定一周内生产的小麦、大豆和糖的数量，以及每个农场各自的总收入。

让x, y{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="X，空格y" style="vertical-align:-6px;height:22px;width:30px" loading="lazy">而且z{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="z" style="vertical-align:-6px;height:22px;width:10px" loading="lazy">分别表示小麦、大豆和糖的价格，单位为每公斤美元。

同一周，农场D的小麦产量是农场a的五分之一，大豆产量是农场B的四分之一，糖产量是农场C的一半。

Grace决定将投资分散到三家公司:公司A、公司B和公司c。她根据风险级别创建了三个不同的投资组合选项，并计算如果一年前投资相同的金额，每个选项今天的价值是多少。

Grace在投资之前听到了A公司增长的好消息，所以她决定把78%的钱投到A公司，剩下的在B公司和C公司之间平均分配。

与前一年相比，公司A未来一年的年度百分比回报率预计将增加26个百分点(因此，如果前一年的回报率为x{"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">%，那么预期的回报是多少(x+26){"language":"en","fontFamily":"Times New Roman","fontSize":"18"}" class="Wirisformula" role="math" alt="左括号x + 26右括号" style="vertical-align:-6px;height:22px;width:58px" loading="lazy">%的未来一年)。对于B公司，预期回报率保持不变，而对于C公司，预期回报率将下降8个百分点。