1. Matrix addition and subtraction1, the arithmetic rule set matrices , , is short, the two matrixes addition and subtraction, i.e. the position of the element addition and subtraction are the same! Note: Only for two matrices with the same number of rows and columns (ie matrix of the same type), addition and subtraction operations are meaningful, that is, addition and subtraction operations are feasible. |
2. The properties of operations (assuming operations are all feasible) satisfy the commutative law and the associative law commutative law ; associative law . |
Second, the multiplication of matrices and numbers1, arithmetic rule number multiplied by the matrix A, the number is multiplied by each element of the matrix A, denoted by or . In particular, it is called the negative matrix. 2. The nature of operation satisfy the associative law and the distribution law the associative law: (λμ)A=λ(μA) ; (λ+μ)A =λA+μA . Distribution law: λ (A+B)=λA+λB . |
Typical examples Example 6.5.1 Known two matrices To satisfy the matrix equation , find the unknown matrix . Solution is known by known conditions |
Three, matrix and matrix multiplication1. Operation rules suppose , then the product of A and B is such a matrix: (1) The number of rows is the same as (left matrix) A, and the number of columns is the same as (right matrix) B, ie . Section (2) C in the first row element of the column by the first A in the first row of elements and B is multiplied by the corresponding column element, then take the sum of the products. |
Typical examples Example 6.5.2 Set matrix Calculate the matrix that the solution is . Let it be think about it : Let the column matrix , the row matrix , and the number of rows and columns respectively be a 3×3 matrix, a 1×1 matrix, that is, there is only one element. |
课堂练习 1、设,,求. 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,行矩阵,求和,比较两个计算结果,能得出什么结论吗? 4、设三阶方阵,三阶单位阵为,试求和,并将计算结果与A比较,看有什么样的结论. |
解: 第1题 . 第2题对于 ,. 求是有意义的,而是无意义的. |
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 是矩阵,是的矩阵. . 结论2 在矩阵的乘法中,必须注意相乘的顺序.即使在与均有意义时,也未必有=成立.可见矩阵乘法不满足交换律. 第4题 计算得:. 结论3 方阵A和它同阶的单位阵作乘积,结果仍为A,即. 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,试计算和. 解 . 结论4 两个非零矩阵的乘积可以是零矩阵.由此若,不能得出或的结论. |
例6.5.4 利用矩阵的乘法,三元线性方程组 可以写成矩阵的形式 = 若记系数、未知量和常数项构成的三个矩阵分别为,,, 则线性方程组又可以简写为矩阵方程的形式:. |
2、 运算性质(假设运算都是可行的) (1) 结合律 . (2) 分配律 (左分配律); (右分配律). (3) . 3、 方阵的幂
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四、矩阵的转置1、 定义
2、运算性质(假设运算都是可行的) (1) (2) (3) (4) ,是常数. |
典型例题 例6.5.5 利用矩阵 验证运算性质: 解 ; 而 所以 . |
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五、方阵的行列式1、定义
2 、运算性质 (1) (行列式的性质) (2) ,特别地: (3) (是常数,A的阶数为n) 思考:设A为阶方阵,那么的行列式与A的行列式之间的关系为什么不是,而是? |
不妨自行设计一个二阶方阵,计算一下和. 例如,则. 于是,而 . 思考:设,有几种方法可以求? 解 方法一:先求矩阵乘法,得到一个二阶方阵,再求其行列式. 方法二:先分别求行列式,再取它们的乘积. |