# Quod Erat Demonstrandum

## 2010/11/04

### 克萊姆法則

Filed under: NSS,Pure Mathematics — johnmayhk @ 6:04 下午
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$\left \{ \begin{array}{ll} ax + by + cz = k_1\\dx + ey + fz = k_2\\gx + hy + iz = k_3\end{array}\right.$

$\Delta = \left|\begin{array}{rcl}a & b & c\\ d & e & f\\ g & h & i\end{array}\right|$

$a,b,c,\dots,i$ 的餘因式（cofactors）分別為 $A,B,C,\dots,I$

$\left \{ \begin{array}{ll} aAx + bAy + cAz = k_1A\\dDx + eDy + fDz = k_2D\\gGx + hGy + iGz = k_3G\end{array}\right.$

$(aA+dD+gG)x + (bA+eD+hG)y + (cA+fD+iG)z = k_1A+k_2D+k_3G$ ………. (*)

$aA+dD+gG = \Delta$
$bA+eD+hG = cA+fD+iG = 0$　［註 1］
$k_1A+k_2D+k_3G = \Delta_x$

$x\Delta = \Delta_x$

(**) $\left \{ \begin{array}{ll} x\Delta = \Delta_x\\y\Delta = \Delta_y\\z\Delta = \Delta_z\end{array}\right.$

$\left \{ \begin{array}{ll} x + 2y + 3z = 1\\2x + 4y + 6z = 2\\3x + 6y + 9z = 5\end{array}\right.$

$\Delta = \Delta_x = \Delta_y = \Delta_z = 0$，但它是無解的。

$\left(\begin{array}{rcl}1 & 2 & 3\\ 2 & 4 & 6\\ 3 & 6 & 9\end{array}\right)$

$\left|\begin{array}{rcl}4 & 6\\ 6 & 9\end{array}\right|$, $-\left|\begin{array}{rcl}2 & 3\\ 6 & 9\end{array}\right|$ 等，都是零。

$\left \{ \begin{array}{ll} aAx + bAy + cAz = k_1A\\dDx + eDy + fDz = k_2D\\gGx + hGy + iGz = k_3G\end{array}\right.$

$\left \{ \begin{array}{ll} x + y = 3\\x - y = 1\end{array}\right.$

$\left \{ \begin{array}{ll} x + y = 3\\2x + 2y = 6\end{array}\right.$

$\left \{ \begin{array}{ll} x + y = 3\\x + y = 4\end{array}\right.$

$\left \{ \begin{array}{ll} 0x + 0y = 0\\ 0x + 0y = 0\end{array}\right.$

$\left \{ \begin{array}{ll} 0x = 0\\ 0y = 0\end{array}\right.$

$x^2 = 9$ 有兩個解（3 和 -3），但我們可以說

「因為 $0x^2 = 0\times 9 = 0$，有無限個解，所以原式 $x^2 = 9$ 都有無限個解」嗎？明顯不能。

［註 1］ 或許同學忘記，略解

$bA+eD+hG = \left|\begin{array}{rcl}b & b & c\\ e & e & f\\ h & h & i\end{array}\right| = 0$

$cA+fD+iG = \left|\begin{array}{rcl}c & b & c\\ f & e & f\\ i & h & i\end{array}\right| = 0$

## 2 則迴響 »

1. 咁如果唔計依一類(三條式都係由一式乘常數而成)既system,
咁4個Delta都係0既時候,可以聲稱system有無限解嗎?

迴響 由 Carmen — 2010/11/04 @ 8:47 下午 | 回應

• Carmen

都是用高斯消去（元）法來判斷方程組的解之情況較佳。

或反問你，比如，除了

$\Delta = \Delta_x = \Delta_y = \Delta_z = 0$ 外，還要加添什麼條件，以確保方程組有解？

迴響 由 johnmayhk — 2010/11/05 @ 12:08 下午 | 回應