克拉默法则
克拉默法则的定义
重要程度：8 分
<h2>克拉默法则的定义</h2>
<p>设 <em>n</em> 元线性方程组为：</p>
<blockquote>
<p>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><br>
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><br>
...<br>
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></p>
</blockquote>
<p>如果系数行列式 <strong>D</strong> = |a<sub>ij</sub>| ≠ 0，则该方程组有唯一解，且解为：</p>
<blockquote>
<p>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</p>
</blockquote>
<p>其中，D<sub>i</sub> 是将系数行列式 <strong>D</strong> 中第 <em>i</em> 列元素替换为常数项 <em>b</em><sub>1</sub>, <em>b</em><sub>2</sub>, ..., <em>b</em><sub>n</sub> 后得到的行列式。</p>
<h2>例题说明</h2>
<h3>例题 1</h3>
<p>求解下列线性方程组：</p>
<blockquote>
<p>2x + 3y = 8<br>
4x + 5y = 14</p>
</blockquote>
<p>解：</p>
<ol>
<li>计算系数行列式 <strong>D</strong>：</li>
<blockquote>
<p>D = |2 3| = 2 * 5 - 3 * 4 = 10 - 12 = -2</p>
<p> |4 5|</p>
</blockquote>
<li>计算 D<sub>1</sub>（将第一列替换为常数项）：</li>
<blockquote>
<p>D<sub>1</sub> = |8 3| = 8 * 5 - 3 * 14 = 40 - 42 = -2</p>
<p> |14 5|</p>
</blockquote>
<li>计算 D<sub>2</sub>（将第二列替换为常数项）：</li>
<blockquote>
<p>D<sub>2</sub> = |2 8| = 2 * 14 - 8 * 4 = 28 - 32 = -4</p>
<p> |4 14|</p>
</blockquote>
<li>根据克拉默法则，解为：</li>
<blockquote>
<p>x = D<sub>1</sub> / D = -2 / -2 = 1</p>
<p>y = D<sub>2</sub> / D = -4 / -2 = 2</p>
</blockquote>
</ol>
<p>因此，方程组的解为：x = 1, y = 2。</p>
<h3>例题 2</h3>
<p>求解下列线性方程组：</p>
<blockquote>
<p>x + 2y + 3z = 6<br>
2x + 5y + 2z = 4<br>
6x - 3y + z = 2</p>
</blockquote>
<p>解：</p>
<ol>
<li>计算系数行列式 <strong>D</strong>：</li>
<blockquote>
<p>D = |1 2 3| = 1(5*1 - 2*(-3)) - 2(2*1 - 2*6) + 3(2*(-3) - 5*6)</p>
<p> |2 5 2| = 1(5 + 6) - 2(2 - 12) + 3(-6 - 30)</p>
<p> |6 -3 1| = 11 + 20 - 108 = -77</p>
</blockquote>
<li>计算 D<sub>1</sub>（将第一列替换为常数项）：</li>
<blockquote>
<p>D<sub>1</sub> = |6 2 3| = 6(5*1 - 2*(-3)) - 2(4*1 - 2*2) + 3(4*(-3) - 5*2)</p>
<p> |4 5 2| = 6(5 + 6) - 2(4 - 4) + 3(-12 - 10)</p>
<p> |2 -3 1| = 66 - 0 - 66 = 0</p>
</blockquote>
<li>计算 D<sub>2</sub>（将第二列替换为常数项）：</li>
<blockquote>
<p>D<sub>2</sub> = |1 6 3| = 1(4*1 - 2*2) - 6(2*1 - 2*6) + 3(2*2 - 4*6)</p>
<p> |2 4 2| = 1(4 - 4) - 6(2 - 12) + 3(4 - 24)</p>
<p> |6 2 1| = 0 + 60 - 60 = 0</p>
</blockquote>
<li>计算 D<sub>3</sub>（将第三列替换为常数项）：</li>
<blockquote>
<p>D<sub>3</sub> = |1 2 6| = 1(5*2 - 4*(-3)) - 2(2*2 - 4*6) + 6(2*(-3) - 5*2)</p>
<p> |2 5 4| = 1(10 + 12) - 2(4 - 24) + 6(-6 - 10)</p>
<p> |6 -3 2| = 22 + 40 - 96 = -34</p>
</blockquote>
<li>根据克拉默法则，解为：</li>
<blockquote>
<p>x = D<sub>1</sub> / D = 0 / -77 = 0</p>
<p>y = D<sub>2</sub> / D = 0 / -77 = 0</p>
<p>z = D<sub>3</sub> / D = -34 / -77 = 34/77</p>
</blockquote>
</ol>
<p>因此，方程组的解为：x = 0, y = 0, z = 34/77。</p>