解题方法
1 . 数值线性代数又称矩阵计算,是计算数学的一个重要分支,其主要研究对象包括向量和矩阵.对于平面向量
,其模定义为
.类似地,对于
行
列的矩阵
,其模可由向量模拓展为
(其中
为矩阵中第
行第
列的数,
为求和符号),记作
,我们称这样的矩阵模为弗罗贝尼乌斯范数,例如对于矩阵
,其矩阵模
.弗罗贝尼乌斯范数在机器学习等前沿领域有重要的应用.
(1)
,
,矩阵
,求使
的
的最小值.
(2)
,
,,矩阵![](https://staticzujuan.xkw.com/quesimg/Upload/formula/880f9f9ab3fcb2dfdfc14d0ab8582fb9.png)
求
.
(3)矩阵
,证明:
,
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/860fc1db2edc066188f8d24e35dbf205.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/332153dce658c8cc26984e355b7c15ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23c529cf68fc1e9a4f9ab4dfbadcfe01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61bb39d4f4036ceed78844592288c408.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37a14c188b1c9d61aa237b137ba18023.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7600d2cfbdc6146db96cc545706004f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e5550d8659980c02488a57afd5964ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/747bedda3150eb258ffb25c923a47614.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c297fac2721a2c7bbaa60b0274dbc34f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de651a4843a0cdbf9e26e51f9c53e837.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83986d667f7618313804888470393a67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c6abaf4851fb819b325eb5d21cd0260.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7013adffb807e769979945ba9aa0809.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83986d667f7618313804888470393a67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/880f9f9ab3fcb2dfdfc14d0ab8582fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f922593fce42b4d7e592e51873aa2ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd8a93d9cf3359a0ad6106ea5360acb.png)
(3)矩阵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c6bed24376a5b1ea247ffb1552eaaf7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83986d667f7618313804888470393a67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62f823d5ffe45a61c388710e7a67fd02.png)
您最近一年使用:0次
2 . 小颖同学在学习探究活动中,定义了一种运等“
”:对于任意实数a,b,都有
,通过研究发现新运算满足交换律:
.小颖提出了两个猜想:
,
,
,①
;②
.
(1)请你任选其中一个猜想,判断其正确与否,若正确,进行证明;若错误,请说明理由;(注:两个猜想都判断、证明或说明理由,仅按第一解答给分)
(2)设
且
,
,当
时,若函数
在区间
上的值域为
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e16415b61722f9961e412386e6819f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4235fe8a6cf0446dbf476822b6dbbce8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac7ff4ffa27279dbf509cfb852446813.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df6593a700bf3e89107556454666b787.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/525c1a68848e95e6b419e0bbec3c0957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11b648347b0e5ed2bdc821dc7cf50d46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c8d87094a7c50f062fa23902cd23c20.png)
(1)请你任选其中一个猜想,判断其正确与否,若正确,进行证明;若错误,请说明理由;(注:两个猜想都判断、证明或说明理由,仅按第一解答给分)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c400a615a16a1662de98dfb4e49d58d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d386474416a278ca29be6075fa076d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49910fc853928999a0acbcc67f4c295c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/733c4ee92975bec9a52b9b2d544d790f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57b85a97933a1d984f6e484b4021c800.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47a964eadb835069b591f479b7c67e8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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2023-12-11更新
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318次组卷
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2卷引用:福建省莆田市莆田第一中学2023-2024学年高一上学期期末考试数学试题