克拉默法则
克拉默法则的应用条件
重要程度：9 分
<h2>克拉默法则的应用条件</h2>
<ul>
<li>方程组必须是线性方程组。</li>
<li>方程组的未知数个数等于方程的个数，即为 <em>n</em> 个未知数和 <em>n</em> 个方程。</li>
<li>系数行列式 <strong>D</strong> 不等于零，即 <strong>D ≠ 0</strong>。</li>
</ul>
<h2>克拉默法则的内容</h2>
<p>对于 <em>n</em> 个未知数 <em>x<sub>1</sub>, x<sub>2</sub>, ..., x<sub>n</sub></em> 的线性方程组：</p>
<p><em>a<sub>11</sub>x<sub>1</sub> + a<sub>12</sub>x<sub>2</sub> + ... + a<sub>1n</sub>x<sub>n</sub> = b<sub>1</sub></em></p>
<p><em>a<sub>21</sub>x<sub>1</sub> + a<sub>22</sub>x<sub>2</sub> + ... + a<sub>2n</sub>x<sub>n</sub> = b<sub>2</sub></em></p>
<p>...</p>
<p><em>a<sub>n1</sub>x<sub>1</sub> + a<sub>n2</sub>x<sub>2</sub> + ... + a<sub>nn</sub>x<sub>n</sub> = b<sub>n</sub></em></p>
<p>如果系数行列式 <strong>D = det(A)</strong> 不等于零，则方程组有唯一解，解为：</p>
<p><em>x<sub>1</sub> = D<sub>1</sub> / D, x<sub>2</sub> = D<sub>2</sub> / D, ..., x<sub>n</sub> = D<sub>n</sub> / D</em></p>
<p>其中，<strong>D<sub>i</sub></strong> 是将系数行列式 <strong>D</strong> 中第 <em>i</em> 列替换为常数项 <em>b<sub>1</sub>, b<sub>2</sub>, ..., b<sub>n</sub></em> 后得到的行列式。</p>
<h2>例题说明</h2>
<h3>例题 1</h3>
<p>解下列线性方程组：</p>
<p><em>2x + 3y = 8</em></p>
<p><em>x - y = 1</em></p>
<h4>解法步骤</h4>
<ol>
<li>写出系数行列式 <strong>D</strong>：</li>
<pre>
D = | 2 3 |
| 1 -1 |
= 2(-1) - 3(1) = -2 - 3 = -5
</pre>
<li>计算 <strong>D<sub>1</sub></strong>（将第一列替换为常数项）：</li>
<pre>
D<sub>1</sub> = | 8 3 |
| 1 -1 |
= 8(-1) - 3(1) = -8 - 3 = -11
</pre>
<li>计算 <strong>D<sub>2</sub></strong>（将第二列替换为常数项）：</li>
<pre>
D<sub>2</sub> = | 2 8 |
| 1 1 |
= 2(1) - 8(1) = 2 - 8 = -6
</pre>
<li>根据克拉默法则，求解 <em>x</em> 和 <em>y</em>：</li>
<pre>
x = D<sub>1</sub> / D = -11 / -5 = 11/5
y = D<sub>2</sub> / D = -6 / -5 = 6/5
</pre>
</ol>
<h3>例题 2</h3>
<p>解下列线性方程组：</p>
<p><em>3x + 2y - z = 1</em></p>
<p><em>2x - 2y + 4z = -2</em></p>
<p><em>-x + 1/2y - z = 0</em></p>
<h4>解法步骤</h4>
<ol>
<li>写出系数行列式 <strong>D</strong>：</li>
<pre>
D = | 3 2 -1 |
| 2 -2 4 |
| -1 1/2 -1 |
= 3((-2)(-1) - (4)(1/2)) - 2((2)(-1) - (4)(-1)) - (-1)((2)(1/2) - (-2)(-1))
= 3(2 - 2) - 2(-2 + 4) + (1 - 2)
= 3(0) - 2(2) + (-1)
= -4 - 1 = -5
</pre>
<li>计算 <strong>D<sub>1</sub></strong>（将第一列替换为常数项）：</li>
<pre>
D<sub>1</sub> = | 1 2 -1 |
| -2 -2 4 |
| 0 1/2 -1 |
= 1((-2)(-1) - (4)(1/2)) - 2((-2)(-1) - (4)(0)) - (-1)((-2)(1/2) - (-2)(0))
= 1(2 - 2) - 2(2) + (1 - 0)
= 1(0) - 4 + 1
= -3
</pre>
<li>计算 <strong>D<sub>2</sub></strong>（将第二列替换为常数项）：</li>
<pre>
D<sub>2</sub> = | 3 1 -1 |
| 2 -2 4 |
| -1 0 -1 |
= 3((-2)(-1) - (4)(0)) - 1((2)(-1) - (4)(-1)) - (-1)((2)(0) - (-2)(-1))
= 3(2) - 1(-2 + 4) + (0 - 2)
= 6 - 2 - 2
= 2
</pre>
<li>计算 <strong>D<sub>3</sub></strong>（将第三列替换为常数项）：</li>
<pre>
D<sub>3</sub> = | 3 2 1 |
| 2 -2 -2 |
| -1 1/2 0 |
= 3((-2)(0) - (-2)(1/2)) - 2((2)(0) - (-2)(-1)) + 1((2)(1/2) - (-2)(-1))
= 3(1) - 2(2) + (1 - 2)
= 3 - 4 - 1
= -2
</pre>
<li>根据克拉默法则，求解 <em>x</em>、<em>y</em> 和 <em>z</em>：</li>
<pre>
x = D<sub>1</sub> / D = -3 / -5 = 3/5
y = D<sub>2</sub> / D = 2 / -5 = -2/5
z = D<sub>3</sub> / D = -2 / -5 = 2/5
</pre>
</ol>