【线性代数】2矩阵
https://i-blog.csdnimg.cn/direct/f898342893f24e0aafef18e0883d977d.png1.矩阵的运算
1.1.界说
矩阵 行列式 数表 数 https://latex.csdn.net/eq?%24A%3D%5Cbegin%7Bpmatrix%7D1%261%26-1%5C%5C2%260%261%5Cend%7Bpmatrix%7D%24 https://latex.csdn.net/eq?%24%7CB%7C%3D%5Cbegin%7Bvmatrix%7D1%261%5C%5C2%260%5Cend%7Bvmatrix%7D%24 行数和列数可以不相称 行数和列数必须相称 1.2.加法与数乘
矩阵的数乘：所有元素都乘这个数
矩阵的加法：对应位置处元素相加
已知https://latex.csdn.net/eq?A%3D%20%5Cbegin%7Bpmatrix%7D%201%20%26%201%20%26%20-1%20%5C%5C%202%20%26%200%20%26%201%20%5Cend%7Bpmatrix%7D%5Cmathrm%7B%2C%7DB%3D%20%5Cbegin%7Bpmatrix%7D%200%20%26%202%20%26%20-1%20%5C%5C%203%20%26%200%20%26%202%20%5Cend%7Bpmatrix%7D，求https://latex.csdn.net/eq?3A+B.
https://latex.csdn.net/eq?%24A%3D%5Cleft%28%5Cbegin%7Barray%7D%7Bccc%7D%5Cfrac%7B1%7D%7B2%7D%26%201%26%20-1%5C%5C0%26%201%26%201%5Cend%7Barray%7D%5Cright%29%24 https://latex.csdn.net/eq?%243A%3D%5Cleft%28%5Cbegin%7Barray%7D%7Bccc%7D3%26%203%26%20-3%5C%5C6%26%200%26%203%5Cend%7Barray%7D%5Cright%29%24 https://latex.csdn.net/eq?%24B%3D%5Cleft%28%5Cbegin%7Barray%7D%7Bccc%7D0%26%202%26%20-1%5C%5C3%26%200%26%202%5Cend%7Barray%7D%5Cright%29%24
https://latex.csdn.net/eq?%243A+B%3D%5Cleft%28%5Cbegin%7Barray%7D%7Bccc%7D3%26%203%26%20-3%5C%5C6%26%200%26%203%5Cend%7Barray%7D%5Cright%29+%5Cleft%28%5Cbegin%7Barray%7D%7Bccc%7D0%26%202%26%20-1%5C%5C3%26%200%26%202%5Cend%7Barray%7D%5Cright%29%3D%5Cleft%28%5Cbegin%7Barray%7D%7Bccc%7D3%26%205%26%20-4%5C%5C9%26%200%26%205%5Cend%7Barray%7D%5Cright%29%24
1.3.乘法
矩阵乘法三步法https://latex.csdn.net/eq?%24A_%7Bm%5Ctimes%20n%7D%2CB_%7Bn%5Ctimes%20k%7D%24
①https://latex.csdn.net/eq?%24AB%24能不能乘：内定乘
②https://latex.csdn.net/eq?%24AB%24乘完是何类型：外定型
③https://latex.csdn.net/eq?%24AB%24中的元素是什么：左出行，右出列
已知https://latex.csdn.net/eq?A%3D%20%5Cbegin%7Bpmatrix%7D%200%20%26%201%20%26%202%20%5C%5C%202%20%26%200%20%26%203%20%5Cend%7Bpmatrix%7D%2CB%3D%20%5Cbegin%7Bpmatrix%7D%201%20%26%201%20%5C%5C%200%20%26%201%20%5C%5C%201%20%26%200%20%5Cend%7Bpmatrix%7D，求https://latex.csdn.net/eq?AB.
由于https://latex.csdn.net/eq?A是两行三列的矩阵，https://latex.csdn.net/eq?B是三行两列的矩阵。以是https://latex.csdn.net/eq?AB是两行两列的矩阵。
https://latex.csdn.net/eq?%5Cbegin%7Baligned%7D%20%5Cmathrm%7BAB%7D%20%26%20%3D%20%5Cbegin%7Bpmatrix%7D%200%20%26%201%20%26%202%20%5C%5C%202%20%26%200%20%26%203%20%5Cend%7Bpmatrix%7D%20%5Cbegin%7Bpmatrix%7D%201%20%26%201%20%5C%5C%200%20%26%201%20%5C%5C%201%20%26%200%20%5Cend%7Bpmatrix%7D%20%5C%5C%20%26%20%3D%20%5Cbegin%7Bpmatrix%7D%200%5Ctimes1+1%5Ctimes0+2%5Ctimes1%20%26%200%5Ctimes1+1%5Ctimes1+2%5Ctimes0%20%5C%5C%202%5Ctimes1+0%5Ctimes0+3%5Ctimes1%20%26%202%5Ctimes1+0%5Ctimes1+3%5Ctimes0%20%5Cend%7Bpmatrix%7D%20%5C%5C%20%26%20%3D%20%5Cbegin%7Bpmatrix%7D%202%20%26%201%20%5C%5C%205%20%26%202%20%5Cend%7Bpmatrix%7D%20%5Cend%7Baligned%7D
1.4.转置
https://latex.csdn.net/eq?%24A%5ET%3A%24 矩阵https://latex.csdn.net/eq?%24A%24的转置，就是将矩阵https://latex.csdn.net/eq?%24A%24行列交换后得到的矩阵。
https://latex.csdn.net/eq?%5Cleft%7CA%5ET%5Cright%7C%3D%5Cleft%7CA%5Cright%7C
已知矩阵https://latex.csdn.net/eq?%24A%3D%5Cbegin%7Bpmatrix%7D2%260%26-1%5C%5C1%263%262%5Cend%7Bpmatrix%7D%24，求矩阵https://latex.csdn.net/eq?%24A%24的转置https://latex.csdn.net/eq?%24A%5ET.%24
矩阵https://latex.csdn.net/eq?A为两行三列的矩阵，https://latex.csdn.net/eq?%24A%3D%5Cbegin%7Bpmatrix%7D2%260%26-1%5C%5C1%263%262%5Cend%7Bpmatrix%7D%24
矩阵https://latex.csdn.net/eq?A%5ET为三行两列的矩阵，https://latex.csdn.net/eq?A%5ET%3D%20%5Cbegin%7Bpmatrix%7D%202%20%26%201%20%5C%5C%200%20%26%203%20%5C%5C%20-1%20%26%202%20%5Cend%7Bpmatrix%7D
1.5.方阵的行列式
方阵行列式的公式：
https://latex.csdn.net/eq?%24%7CkA%7C%3Dk%5En%5Cleft%7CA%5Cright%7C%24
https://latex.csdn.net/eq?%24%5Cleft%7CA%5Cboldsymbol%7BB%7D%5Cright%7C%3D%5Cleft%7C%5Cboldsymbol%7BB%7DA%5Cright%7C%3D%5Cleft%7CA%5Cright%7C%5Cleft%7C%5Cboldsymbol%7BB%7D%5Cright%7C%24
设https://latex.csdn.net/eq?A%2CB为四阶方阵，且https://latex.csdn.net/eq?%7CA%7C%3D3，且https://latex.csdn.net/eq?%7C3B%7C%3D81，https://latex.csdn.net/eq?%7CAB%7C%3D________.
https://latex.csdn.net/eq?%7C3B%7C%20%3D%203%5E4%20%7CB%7C%5C%5C%20%7C3B%7C%20%3D%2081%20%7CB%7C%20%3D%2081%5C%5C%20%7CB%7C%20%3D%201%20%5C%5C%7CA%7C%20%3D%203%5C%5C%20%7CAB%7C%20%3D%20%7CA%7C%20%7CB%7C%20%3D%203%5Ctimes%201%20%3D%203
<hr> 设https://latex.csdn.net/eq?A%2CB为三阶方阵，且https://latex.csdn.net/eq?%7CA%7C%3D-3%2C%7CB%7C%3D2%2C则https://latex.csdn.net/eq?%7C2A%5ETB%7C%3D________.
根据公式https://latex.csdn.net/eq?%7CkA%7C%3Dk%5En%7CA%7C可得https://latex.csdn.net/eq?%7C2A%5ETB%7C%3D8%7CA%5ETB%7C。
根据公式https://latex.csdn.net/eq?%7CAB%7C%3D%7CA%7C%7CB%7C%2C%7CA%5ET%7C%3D%7CA%7C可得https://latex.csdn.net/eq?%7CA%5ETB%7C%3D%7CA%7C%7CB%7C。
以是https://latex.csdn.net/eq?%7C2A%5ETB%7C%3D8%7CA%7C%7CB%7C%3D8%5Ctimes%20%28-3%29%5Ctimes2%3D-48
2.矩阵的求逆
逆矩阵就相称于矩阵的“倒数”，对于一个方阵(行数和列数相称的矩阵)https://latex.csdn.net/eq?%24A%24，如果存在另一个矩阵https://latex.csdn.net/eq?%24B%24，使得https://latex.csdn.net/eq?%24A%24乘以https://latex.csdn.net/eq?%24B%24即是单位矩阵https://latex.csdn.net/eq?%24E%24 (单位矩阵就像数字中的https://latex.csdn.net/eq?1，主对角线上都是https://latex.csdn.net/eq?1，其他位置都是https://latex.csdn.net/eq?0)，那么https://latex.csdn.net/eq?%24B%24就是https://latex.csdn.net/eq?%24A%24的逆矩阵，记作https://latex.csdn.net/eq?%24A%5E%7B-1%7D%24。
可逆的充要条件
矩阵https://latex.csdn.net/eq?A可逆的充要条件是https://latex.csdn.net/eq?%7CA%7C%5Cneq0
已知矩阵https://latex.csdn.net/eq?%24A%3D%5Cbegin%7Bpmatrix%7D1%261%263%5C%5C0%262%262%5C%5C0%260%26-1%5Cend%7Bpmatrix%7D%24，判断矩阵https://latex.csdn.net/eq?%24A%24是否为可逆矩阵。
由于https://latex.csdn.net/eq?%24%7CA%7C%24为上三角形行列式，以是https://latex.csdn.net/eq?%24%7CA%7C%24即是主对角线元素相乘
即https://latex.csdn.net/eq?%24%7CA%7C%3D-2%24
由于https://latex.csdn.net/eq?%24%7CA%7C%5Cneq0%24，以是矩阵https://latex.csdn.net/eq?%24A%24可逆
2.1.陪同矩阵法（二阶具体方阵求逆）
陪同矩阵
陪同矩阵主要用于求逆矩阵，因为有一个重要的公式https://latex.csdn.net/eq?%24A%5E%7B-%201%7D%3D%20%5Cfrac%20%7BA%5E*%20%7D%7B%7C%20A%7C%20%7D%24，此中https://latex.csdn.net/eq?%24%7CA%7C%24是矩阵https://latex.csdn.net/eq?%24A%24的行列式。当https://latex.csdn.net/eq?%24%7CA%7C%5Cneq0%24时，就可以通过这个公式利用陪同矩
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