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# 矩阵行运算

## 交换任意两行

### 例题

$\left[\begin{array}{rrr}4& 8& 3\\ 2& 4& 5\\ 7& 1& 2\end{array}\right]$

### 解法

${R}_{1}↔{R}_{2}$ 表示将第 $1$ 行和第 $2$ 行互换。

$\left[\begin{array}{rrr}4& 8& 3\\ 2& 4& 5\\ 7& 1& 2\end{array}\right]\stackrel{{R}_{1}↔{R}_{2}\phantom{\rule[-.25em]{0ex}{0ex}}}{\to }\left[\begin{array}{rrr}2& 4& 5\\ 4& 8& 3\\ 7& 1& 2\end{array}\right]$

$\left[\begin{array}{rrr}7& 2& 9\\ 6& 4& 1\\ 1& 3& 12\end{array}\right]$

## 将一行乘以非零的常数

### 例题

$\left[\begin{array}{rrr}6& 6& 1\\ 2& 3& 0\\ 4& 5& 9\end{array}\right]$

### 解法

$3{R}_{2}\to {R}_{2}$ 表示把第 $2$ 行变成 $3$ 乘以自身。
$\left[\begin{array}{rrr}6& 6& 1\\ 2& 3& 0\\ 4& 5& 9\end{array}\right]$ 变成 $\left[\begin{array}{rrr}6& 6& 1\\ 3\cdot 2& 3\cdot 3& 3\cdot 0\\ 4& 5& 9\end{array}\right]=\left[\begin{array}{rrr}6& 6& 1\\ 6& 9& 0\\ 4& 5& 9\end{array}\right]$

$\left[\begin{array}{rrr}6& 6& 1\\ 2& 3& 0\\ 4& 5& 9\end{array}\right]\stackrel{3{R}_{2}\to {R}_{2}\phantom{\rule[-.25em]{0ex}{0ex}}}{\to }\left[\begin{array}{rrr}6& 6& 1\\ 6& 9& 0\\ 4& 5& 9\end{array}\right]$

$\left[\begin{array}{cccc}2& 6& 5& 1\\ 7& 4& 8& 0\end{array}\right]$

## 将一行与另一行相加

### 例题

$\left[\begin{array}{rrr}2& 3& 4\\ 0& 8& 1\end{array}\right]$

### 解法

${R}_{1}+{R}_{2}\to {R}_{2}$ 表示把第 $2$ 行变成第 $1$ 行和第 $2$ 行的和。
$\left[\begin{array}{rrr}2& 3& 4\\ 0& 8& 1\end{array}\right]$ 变成 $\left[\begin{array}{lll}2& 3& 4\\ 2+0& 3+8& 4+1\end{array}\right]=\left[\begin{array}{rrr}2& 3& 4\\ 2& 11& 5\end{array}\right]$

$\left[\begin{array}{rrr}2& 3& 4\\ 0& 8& 1\end{array}\right]\stackrel{{R}_{1}+{R}_{2}\to {R}_{2}\phantom{\rule[-.25em]{0ex}{0ex}}}{\to }\left[\begin{array}{rrr}2& 3& 4\\ 2& 11& 5\end{array}\right]$

$\left[\begin{array}{rrr}-1& 6& -2\\ -3& 5& 0\\ 7& 2& 1\end{array}\right]$

$\left[\begin{array}{rrr}-5& 7& 3\\ -2& -1& 4\\ 8& 8& -6\end{array}\right]$

## 方程组和矩阵行的计算

$\begin{array}{rl}1x+3y& =5\\ 2x+5y& =6\end{array}$$\left[\begin{array}{ccc}1& 3& 5\\ 2& 5& 6\end{array}\right]$

### 交换任意两行

$\begin{array}{rl}1x+3y& =5\\ 2x+5y& =6\end{array}$$\left[\begin{array}{ccc}1& 3& 5\\ \\ 2& 5& 6\end{array}\right]$
$↓$
$\begin{array}{rl}2x+5y& =6\\ 1x+3y& =5\end{array}$$\left[\begin{array}{ccc}2& 5& 6\\ \\ 1& 3& 5\end{array}\right]$

### 将一行乘以非零的常数

$\begin{array}{rl}1x+3y& =5\\ 2x+5y& =6\end{array}$$\left[\begin{array}{ccc}1& 3& 5\\ 2& 5& 6\end{array}\right]$
$↓$
$\begin{array}{rl}-2x+\left(-6\right)y& =-10\\ 2x+\phantom{\left(-\right)}5y& =6\end{array}$$\left[\begin{array}{rrr}-2& -6& -10\\ 2& 5& 6\end{array}\right]$

### 将一行与另一行相加

$\begin{array}{rl}-2x-6y& =-10\\ 2x+5y& =6\end{array}$$\left[\begin{array}{rrr}-2& -6& -10\\ 2& 5& 6\end{array}\right]$
$↓$
$\begin{array}{rl}-2x+\left(-6\right)y& =-10\\ 0x+\left(-1\right)y& =-4\end{array}$$\left[\begin{array}{rrr}-2& -6& -10\\ 0& -1& -4\end{array}\right]$