今天清北学堂信息学金牌教研团队给大家汇总了一下矩阵的运算

一、矩阵的加法与减法

1、运算规则　　设矩阵

https://5b0988e595225.cdn.sohucs.com/images/20180329/70642bc54e2546e8a3cf47275ff2f32d.jpg

则

https://5b0988e595225.cdn.sohucs.com/images/20180329/87a502e84ff44f0b90f677d6fb98b5b1.jpg

清北学堂信息学金牌教研团队提醒，两个矩阵相加减，即它们相同位置的元素相加减！注意：只有对于两个行数、列数分别相等的矩阵（即同型矩阵），加减法运算才有意义，即加减运算是可行的．　　

2、运算性质（假设运算都是可行的）　　

满足交换律和结合律

　　交换律　

https://5b0988e595225.cdn.sohucs.com/images/20180329/29c7db1d802e408e8eaf779649c1098d.jpg

；

　　结合律　

https://5b0988e595225.cdn.sohucs.com/images/20180329/39eed88f770f47e6a45b3fa116e97086.jpg

．

二、矩阵与数的乘法

　　1、运算规则

数

https://5b0988e595225.cdn.sohucs.com/images/20180329/8d4770f5d85d4f6aad7d6e2fd369b173.jpg

乘矩阵A，就是将数

https://5b0988e595225.cdn.sohucs.com/images/20180329/8d4770f5d85d4f6aad7d6e2fd369b173.jpg

乘矩阵A中的每一个元素，记为

https://5b0988e595225.cdn.sohucs.com/images/20180329/6326d73d0a414ebb89e8c9f1f3f84f98.jpg

或

https://5b0988e595225.cdn.sohucs.com/images/20180329/10547f3457f54364bc47a8bcc6afd246.jpg

．　　特别地，称

https://5b0988e595225.cdn.sohucs.com/images/20180329/b45fec4f56554cafb90eb2a80fc2ab10.jpg

称为

https://5b0988e595225.cdn.sohucs.com/images/20180329/997a678b6589470994bd0d72f5b5ed95.jpg

的负矩阵．　　

2、运算性质　　满足结合律和分配律　　结合律：(λμ)A=λ(μA)；(λ+μ)A =λA+μA．　　分配律：λ(A+B)=λA+λB．典型例题例6.5.1　已知两个矩阵

https://5b0988e595225.cdn.sohucs.com/images/20180329/b750690c4b23440993fad5949650e656.jpg

满足矩阵方程

https://5b0988e595225.cdn.sohucs.com/images/20180329/a297a4fbac90442688a2f115c0f0f0ff.jpg

，求未知矩阵

https://5b0988e595225.cdn.sohucs.com/images/20180329/2c98a894116c498abbb3f1f165ff4b85.jpg

．解　由已知条件知

https://5b0988e595225.cdn.sohucs.com/images/20180329/fd882dc74f544ad78941a97c420a9a0b.jpg

https://5b0988e595225.cdn.sohucs.com/images/20180329/a50ab583c8b74142b59670769bcf2e2f.jpg

三、矩阵与矩阵的乘法

　　1、运算规则　　设

https://5b0988e595225.cdn.sohucs.com/images/20180329/997a678b6589470994bd0d72f5b5ed95.jpg

，

https://5b0988e595225.cdn.sohucs.com/images/20180329/9bf4e7ec435d4399a8926051b7415fde.jpg

，则A与B的乘积

https://5b0988e595225.cdn.sohucs.com/images/20180329/2c4f75ab5e0447a4bc999a6205e0af79.jpg

是这样一个矩阵：　　(1) 行数与（左矩阵）A相同，列数与（右矩阵）B相同，即

https://5b0988e595225.cdn.sohucs.com/images/20180329/fde91d2450f74845bfd518a5d61e3755.jpg

．　　(2) C的第

https://5b0988e595225.cdn.sohucs.com/images/20180329/d9430f19d01142c7bae33ce012926f4b.jpg

行第

https://5b0988e595225.cdn.sohucs.com/images/20180329/e4db04d751754537ac3ebbcb81d39740.jpg

列的元素

https://5b0988e595225.cdn.sohucs.com/images/20180329/5fc4dd6bcf444e80ad96a062ca844705.jpg

由A的第

https://5b0988e595225.cdn.sohucs.com/images/20180329/d9430f19d01142c7bae33ce012926f4b.jpg

行元素与B的第

https://5b0988e595225.cdn.sohucs.com/images/20180329/e4db04d751754537ac3ebbcb81d39740.jpg

列元素对应相乘，再取乘积之和．典型例题例6.5.2　设矩阵

https://5b0988e595225.cdn.sohucs.com/images/20180329/aec53f6b5df3470594205c78c7b3e547.jpg

计算

https://5b0988e595225.cdn.sohucs.com/images/20180329/55742348b44a437ca1bd77dc533bbb7c.jpg

解

https://5b0988e595225.cdn.sohucs.com/images/20180329/c8a7ab4c1043465ca39f47a653b2d26a.jpg

是

https://5b0988e595225.cdn.sohucs.com/images/20180329/49010d38249549cd89a271ec72940a63.jpg

的矩阵．设它为

https://5b0988e595225.cdn.sohucs.com/images/20180329/9844fb87143b4ab4abf44a172e45106e.jpg

https://5b0988e595225.cdn.sohucs.com/images/20180329/142e247091d44be581cae9f928ff29b4.jpg

https://5b0988e595225.cdn.sohucs.com/images/20180329/7146c1529bb24215ab74e262431cc6d4.jpg

https://5b0988e595225.cdn.sohucs.com/images/20180329/100bf2313d014bcab642466ad448245b.jpg

https://5b0988e595225.cdn.sohucs.com/images/20180329/9844fb87143b4ab4abf44a172e45106e.jpg

https://5b0988e595225.cdn.sohucs.com/images/20180329/142e247091d44be581cae9f928ff29b4.jpg

https://5b0988e595225.cdn.sohucs.com/images/20180329/7146c1529bb24215ab74e262431cc6d4.jpg

https://5b0988e595225.cdn.sohucs.com/images/20180329/633e252970474671abb68c7073668d67.jpg

想一想：设列矩阵

https://5b0988e595225.cdn.sohucs.com/images/20180329/150aecacc5a04bc982993352757008a7.jpg

，行矩阵

https://5b0988e595225.cdn.sohucs.com/images/20180329/01a2ceab75f8429287662c9323bada39.jpg

，

https://5b0988e595225.cdn.sohucs.com/images/20180329/c8a7ab4c1043465ca39f47a653b2d26a.jpg

和

https://5b0988e595225.cdn.sohucs.com/images/20180329/dc2e8ef6623b45cab353bf5eff85b67e.jpg

的行数和列数分别是多少呢

https://5b0988e595225.cdn.sohucs.com/images/20180329/33c45ca7a96b401798dc9ed852ad6ac1.jpg

https://5b0988e595225.cdn.sohucs.com/images/20180329/c8a7ab4c1043465ca39f47a653b2d26a.jpg

是3×3的矩阵，

https://5b0988e595225.cdn.sohucs.com/images/20180329/dc2e8ef6623b45cab353bf5eff85b67e.jpg

是1×1的矩阵，即

https://5b0988e595225.cdn.sohucs.com/images/20180329/dc2e8ef6623b45cab353bf5eff85b67e.jpg

只有一个元素．课堂练习　　1、设

https://5b0988e595225.cdn.sohucs.com/images/20180329/0a74c371ecd04cefada16b27179eb60b.jpg

，

https://5b0988e595225.cdn.sohucs.com/images/20180329/b2d4eb11b4f14512bb95a7ae83873f8d.jpg

，求

https://5b0988e595225.cdn.sohucs.com/images/20180329/c8a7ab4c1043465ca39f47a653b2d26a.jpg

．　　

2、在第1道练习题中，两个矩阵相乘的顺序是A在左边，B在右边，称为A左乘B或B右乘A．如果交换顺序，让B在左边，A在右边，即A右乘B，运算还能进行吗？请算算试试看．并由此思考：两个矩阵应当满足什么条件，才能够做乘法运算．　　

3、设列矩阵

https://5b0988e595225.cdn.sohucs.com/images/20180329/150aecacc5a04bc982993352757008a7.jpg

，行矩阵

https://5b0988e595225.cdn.sohucs.com/images/20180329/01a2ceab75f8429287662c9323bada39.jpg

，求

https://5b0988e595225.cdn.sohucs.com/images/20180329/c8a7ab4c1043465ca39f47a653b2d26a.jpg

和

https://5b0988e595225.cdn.sohucs.com/images/20180329/dc2e8ef6623b45cab353bf5eff85b67e.jpg

，比较两个计算结果，能得出什么结论吗？　　

4、设三阶方阵

https://5b0988e595225.cdn.sohucs.com/images/20180329/31d79242d02041489a78d848668ecbbb.jpg

，三阶单位阵为

https://5b0988e595225.cdn.sohucs.com/images/20180329/5f5a3c5679c44c72aac5423af2e6f11f.jpg

，试求https://5b0988e595225.cdn.sohucs.com/images/20180329/9a40afb22c0d47319a402908930dc75e.jpg，并将计算结果与A比较，看有什么样的结论．

解：　　第1题

https://5b0988e595225.cdn.sohucs.com/images/20180329/23e89555408d480296b4fe95cb2e0196.jpg

．　　第2题　　对于

https://5b0988e595225.cdn.sohucs.com/images/20180329/0a74c371ecd04cefada16b27179eb60b.jpg

，

https://5b0988e595225.cdn.sohucs.com/images/20180329/b2d4eb11b4f14512bb95a7ae83873f8d.jpg

．　　求https://5b0988e595225.cdn.sohucs.com/images/20180329/c8a7ab4c1043465ca39f47a653b2d26a.jpg是有意义的，而是无意义的．

清北学堂信息学金牌教研团队结论

结论1　只有在下列情况下，两个矩阵的乘法才有意义，或说乘法运算是可行的：左矩阵的列数＝右矩阵的行数．　　

第3题

https://5b0988e595225.cdn.sohucs.com/images/20180329/c8a7ab4c1043465ca39f47a653b2d26a.jpg是矩阵，是的矩阵．

https://5b0988e595225.cdn.sohucs.com/images/20180329/2efaff237f6e46429d8c5d9a8b13f8e3.jpg

https://5b0988e595225.cdn.sohucs.com/images/20180329/b83163617c4e47f9816b3be1eea1377e.jpg

．

https://5b0988e595225.cdn.sohucs.com/images/20180329/f0469901b5b94f18aca510a5a07bc56e.jpg

https://5b0988e595225.cdn.sohucs.com/images/20180329/ac890bb534914129b89fe47921ebe6de.jpg

结论2　在矩阵的乘法中，必须注意相乘的顺序．即使在https://5b0988e595225.cdn.sohucs.com/images/20180329/b74c649ce3624d36874d0d8ca0f73060.jpg成立．可见矩阵乘法不满足交换律．　　

第4题　　计算得：https://5b0988e595225.cdn.sohucs.com/images/20180329/97f8c30036b34d1e9e78d3f7b56270ae.jpg

．　　

结论3　方阵A和它同阶的单位阵作乘积，结果仍为A，即

https://5b0988e595225.cdn.sohucs.com/images/20180329/97f8c30036b34d1e9e78d3f7b56270ae.jpg

．　　单位阵在矩阵乘法中的作用相当于数1在我们普通乘法中的作用．典型例题例6.5.3　设

https://5b0988e595225.cdn.sohucs.com/images/20180329/dc2e8ef6623b45cab353bf5eff85b67e.jpg

．解

https://5b0988e595225.cdn.sohucs.com/images/20180329/23472de7e2c84eaf9283e86c64178054.jpg

https://5b0988e595225.cdn.sohucs.com/images/20180329/be13d4fa3f92427593f0cfdff33574df.jpg

https://5b0988e595225.cdn.sohucs.com/images/20180329/ba66dd894a5541f3a2d06616abd914d0.jpg

．

https://5b0988e595225.cdn.sohucs.com/images/20180329/e970d34f438e47ebb0e224a73ed6e929.jpg

https://5b0988e595225.cdn.sohucs.com/images/20180329/bd10bc467ff04ef1bdef9a9a34b8ac10.jpg

https://5b0988e595225.cdn.sohucs.com/images/20180329/881b54b0488b402682806a2c1a2c9745.jpg

结论4　两个非零矩阵的乘积可以是零矩阵．由此若https://5b0988e595225.cdn.sohucs.com/images/20180329/8d328e50f0954661bf05a1bfc545ab65.jpg的结论．

例6.5.4　利用矩阵的乘法，三元线性方程组

https://5b0988e595225.cdn.sohucs.com/images/20180329/6ec978f9ae94425894bdb0fff5cdd644.jpg

可以写成矩阵的形式

https://5b0988e595225.cdn.sohucs.com/images/20180329/afac16add7a5482cac54b045d8fc22a4.jpg

https://5b0988e595225.cdn.sohucs.com/images/20180329/3c98a3f4c37b45e592ad6da51ecec5e6.jpg

＝

https://5b0988e595225.cdn.sohucs.com/images/20180329/a6ea9e91904443f191df7408a8716905.jpg

若记系数、未知量和常数项构成的三个矩阵分别为

https://5b0988e595225.cdn.sohucs.com/images/20180329/3898d92b98b0475186b297d1fbb33f5a.jpg

https://5b0988e595225.cdn.sohucs.com/images/20180329/afac16add7a5482cac54b045d8fc22a4.jpg

，

https://5b0988e595225.cdn.sohucs.com/images/20180329/b3f214030148410ca3337228d3716995.jpg

，

https://5b0988e595225.cdn.sohucs.com/images/20180329/26109dd0102a4491a06072eb44009016.jpg

，　　则线性方程组又可以简写为矩阵方程的形式：https://5b0988e595225.cdn.sohucs.com/images/20180329/d2febb0a155f45e4a68cb4547fb214cb.jpg

2、运算性质（假设运算都是可行的）　　

(1)　结合律　

https://5b0988e595225.cdn.sohucs.com/images/20180329/1d45802c40a14a1d94e31db900280b67.jpg

．　　(2)　分配律　

https://5b0988e595225.cdn.sohucs.com/images/20180329/f58ffb22ed004dedb70cd4f293af9f4b.jpg

（左分配律）；

https://5b0988e595225.cdn.sohucs.com/images/20180329/307db16a84784172ac40148e8c61d186.jpg

（右分配律）．　　(3)　

https://5b0988e595225.cdn.sohucs.com/images/20180329/a439a1b26ea740e083eb8412ef20e45a.jpg

3、方阵的幂定义：设A是方阵，https://5b0988e595225.cdn.sohucs.com/images/20180329/fba43933bd17470fa345c671c98971bf.jpg，

https://5b0988e595225.cdn.sohucs.com/images/20180329/0960f77f98404a79a5d78e3ee21fda7f.jpg

显然，记号https://5b0988e595225.cdn.sohucs.com/images/20180329/b04b645a0707496d9f782999e9ecf700.jpg个A的连乘积．

下面是有清北学堂信息学金牌教研团队给大家总结的矩阵的转置

四、矩阵的转置

1、定义

定义：将矩阵A的行换成同序号的列所得到的新矩阵称为矩阵A的转置矩阵，记作

https://5b0988e595225.cdn.sohucs.com/images/20180329/0ac77b7dba2043069b1662148c61db50.jpg

或

https://5b0988e595225.cdn.sohucs.com/images/20180329/8543f8e002fe48a4bcadc57af10d91be.jpg

．　　例如，矩阵

https://5b0988e595225.cdn.sohucs.com/images/20180329/0a74c371ecd04cefada16b27179eb60b.jpg

的转置矩阵为

https://5b0988e595225.cdn.sohucs.com/images/20180329/b642d99cfaf84feeb851cede82f13e80.jpg

．

　　2、运算性质（假设运算都是可行的）　

　(1)　https://5b0988e595225.cdn.sohucs.com/images/20180329/e1700f17fcc44295b96160232d1e830f.jpg

　　(2)　https://5b0988e595225.cdn.sohucs.com/images/20180329/8c0b427ea0a34f179f87c8825690957d.jpg

　　(3)　

https://5b0988e595225.cdn.sohucs.com/images/20180329/f594de0b051846ca9492ed6726dd0242.jpg

　　(4)　

https://5b0988e595225.cdn.sohucs.com/images/20180329/961dc80356754d9990d4924ab6799a52.jpg

，

https://5b0988e595225.cdn.sohucs.com/images/20180329/f24f95f9156c4e029aa0f4abd84d874f.jpg

是常数．

　　2、运算性质（假设运算都是可行的）　

(1)　https://5b0988e595225.cdn.sohucs.com/images/20180329/e1700f17fcc44295b96160232d1e830f.jpg

　　(2)　

https://5b0988e595225.cdn.sohucs.com/images/20180329/8c0b427ea0a34f179f87c8825690957d.jpg

　　(3)　

https://5b0988e595225.cdn.sohucs.com/images/20180329/f594de0b051846ca9492ed6726dd0242.jpg

　　(4)　

https://5b0988e595225.cdn.sohucs.com/images/20180329/961dc80356754d9990d4924ab6799a52.jpg

，

https://5b0988e595225.cdn.sohucs.com/images/20180329/f24f95f9156c4e029aa0f4abd84d874f.jpg

是常数．典型例题 例6.5.5 利用矩阵

https://5b0988e595225.cdn.sohucs.com/images/20180329/1b8ebbf1774d4962b5cb7babd55c7f79.jpg

验证运算性质：

https://5b0988e595225.cdn.sohucs.com/images/20180329/0b6badb20004470ba0ffb52989b7246b.jpg

解

https://5b0988e595225.cdn.sohucs.com/images/20180329/6f3459569c244f75a073d4f34e32d00d.jpg

https://5b0988e595225.cdn.sohucs.com/images/20180329/dc77e1c48b7c403a95043b52b0532473.jpg

https://5b0988e595225.cdn.sohucs.com/images/20180329/9af76a91717348ed91d8dd49f726082c.jpg

而

https://5b0988e595225.cdn.sohucs.com/images/20180329/8ebdf47892a148b5830ac853518a106c.jpg

所以

https://5b0988e595225.cdn.sohucs.com/images/20180329/8501300afff247718f67768ee8f9b9d8.jpg

．

定义：如果方阵满足https://5b0988e595225.cdn.sohucs.com/images/20180329/88f61a4a0a234978836d40d8f323b0e8.jpg，则称A为对称矩阵．

对称矩阵的特点是：它的元素以主对角线为对称轴对应相等．

五、方阵的行列式

1、定义

定义：由方阵A的元素所构成的行列式（各元素的位置不变），称为方阵A的行列式，记作

https://5b0988e595225.cdn.sohucs.com/images/20180329/ff5b0b3fa00e4456b2e22c5cd537a7a0.jpg

或

https://5b0988e595225.cdn.sohucs.com/images/20180329/6bf75d43f3a04929a3a4ddd9ecaab996.jpg

2、运算性质　　(1)

https://5b0988e595225.cdn.sohucs.com/images/20180329/171dc75fd564406a8243d7cfe25d4b2b.jpg

（行列式的性质）　　(2)

https://5b0988e595225.cdn.sohucs.com/images/20180329/9bf786bf1e134bfbb5638b6573b6c3e2.jpg

，特别地：

https://5b0988e595225.cdn.sohucs.com/images/20180329/80ad40aa56554cf3924c7d6879fd705e.jpg

(3)

https://5b0988e595225.cdn.sohucs.com/images/20180329/1867e7db860b467383a7a8d2a1665a2a.jpg

（

https://5b0988e595225.cdn.sohucs.com/images/20180329/f24f95f9156c4e029aa0f4abd84d874f.jpg

是常数，A的阶数为n）思考：设A为

https://5b0988e595225.cdn.sohucs.com/images/20180329/aca2d92fcbce49d49d4013ca1608c3bc.jpg

阶方阵，那么

https://5b0988e595225.cdn.sohucs.com/images/20180329/61f62c837b104d6da8e29167a06e1ef6.jpg

的行列式

https://5b0988e595225.cdn.sohucs.com/images/20180329/fbfc89b61b2a49928359b6bfcb1c0417.jpg

与A的行列式

https://5b0988e595225.cdn.sohucs.com/images/20180329/ff5b0b3fa00e4456b2e22c5cd537a7a0.jpg

之间的关系为什么不是

https://5b0988e595225.cdn.sohucs.com/images/20180329/af9131d92c8846c39156ccd927ccf69f.jpg

，而是

https://5b0988e595225.cdn.sohucs.com/images/20180329/1867e7db860b467383a7a8d2a1665a2a.jpg

？　　不妨自行设计一个二阶方阵，计算一下

https://5b0988e595225.cdn.sohucs.com/images/20180329/a8d8c3c24e544377935e4e0e0b01658f.jpg

和

https://5b0988e595225.cdn.sohucs.com/images/20180329/53385859ad204b2ea2e0976691f75e3b.jpg

．　　例如

https://5b0988e595225.cdn.sohucs.com/images/20180329/0b0332b9a2674063aef332ca7c8be781.jpg

，则

https://5b0988e595225.cdn.sohucs.com/images/20180329/5ebf46d63bd740159a43eb5cf841c537.jpg

．　　于是

https://5b0988e595225.cdn.sohucs.com/images/20180329/56f68cd044cb4b4e8caab5c30b260aef.jpg

，而

https://5b0988e595225.cdn.sohucs.com/images/20180329/4e88b588e77f4c5b86e0a064ef8a2fb8.jpg

https://5b0988e595225.cdn.sohucs.com/images/20180329/058556ce573148c9812d5b9c613175ec.jpg

．思考：设

https://5b0988e595225.cdn.sohucs.com/images/20180329/345c9b102ef94caaa94b3f87e2e9e570.jpg

，有几种方法可以求

https://5b0988e595225.cdn.sohucs.com/images/20180329/af149bd0e434425a8150d86f626514e4.jpg

？解 方法一：先求矩阵乘法

https://5b0988e595225.cdn.sohucs.com/images/20180329/c8a7ab4c1043465ca39f47a653b2d26a.jpg

，得到一个二阶方阵，再求其行列式．　　　　方法二：先分别求行列式

https://5b0988e595225.cdn.sohucs.com/images/20180329/9bb5546df6dc4db2b7fd64e90795cad4.jpg，再取它们的乘积．

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