二階反方陣
Author: 藍邦偉  老師

二階反方陣

A^{-1}=\frac{1}{det(A) }\left [ \begin{array}{} d & -b \ -c & a\ \end{array}  \right ] 

a=

b=

c=

d=

A=\left [ \begin{array}{} a & b \ c & d \ \end{array}  \right ] 

a=

b=

c=

d=

$第$%v$步$

$ ∵AA^{-1}=I $

1

2

$⇒ \Delta=0\;,\;反方陣不存在 $

$第$%v$步$

$ 公式解 $

$ 定義 $

s = 2 (slider from 0 to 5, step 0.1)
L1 = Sequence(Random[-5, 5] + s * 0, i, 1, 20)
a = 1
d = 4
c = 3
b = 2
det: a * d - b * c
inputBox1 = Textfield(a)
inputBox2 = Textfield(b)
inputBox3 = Textfield(c)
inputBox4 = Textfield(d)
text1: "A=\left [ \begin{array}{}" + (LaTeX[a]) + " & " + (LaTeX[b]) + " \\ " + (LaTeX[c]) + " &" + (LaTeX[d]) + " \\ \end{array}  \right ] ⇒det(A)=\left\vert\begin{array}{cc} " + (LaTeX[a]) + " & " + (LaTeX[b]) + " \\ " + (LaTeX[c]) + " & " + (LaTeX[d]) + " \\ \end{array}   \right\vert =" + (LaTeX[det]) + ""
text2: "$ A^{-1}=" + (LaTeX[FractionText[1 / det]]) + "\left [ \begin{array}{cc} " + (LaTeX[d]) + " & " + (LaTeX[-b]) + " \\ " + (LaTeX[-c]) + " & " + (LaTeX[a]) + " \\ \end{array}  \right ] =\left [ \begin{array}{cc} " + (LaTeX[FractionText[d / det]]) + " & " + (LaTeX[FractionText[(-b) / det]]) + " \\ " + (LaTeX[FractionText[(-c) / det]]) + " & " + (LaTeX[FractionText[a / det]]) + " \\ \end{array}  \right ]  $"
n = 0 (slider from 0 to 2, step 1)
T3: "$ ∴\left [ \begin{array}{cc} " + (LaTeX[a]) + " & " + (LaTeX[b]) + " \\ " + (LaTeX[c]) + " & " + (LaTeX[d]) + " \\ \end{array}  \right ] \left [ \begin{array}{cc} x & y \\ z & u \\ \end{array}  \right ]=\left [ \begin{array}{cc} 1 & 0 \\0 & 1 \\ \end{array}  \right ]   $"
T4: "$ \left [\begin{array}{cc} " + (LaTeX[a]) + "x" + (LaTeX[If[b < 0, "-", "+"]]) + "" + (LaTeX[abs(b)]) + "z & " + (LaTeX[a]) + "y" + (LaTeX[If[b < 0, "-", "+"]]) + "" + (LaTeX[abs(b)]) + "u \\ " + (LaTeX[c]) + "x" + (LaTeX[If[d < 0, "-", "+"]]) + "" + (LaTeX[abs(d)]) + "z & " + (LaTeX[c]) + "y" + (LaTeX[If[d < 0, "-", "+"]]) + "" + (LaTeX[abs(d)]) + "u \\ \end{array}   \right ] =\left [\begin{array}{cc} 1 & 0 \\ 0 & 1\\ \end{array}   \right ]  $"
T5: "$ \left\lbrace\begin{array}{ll} " + (LaTeX[a]) + "x" + (LaTeX[If[b < 0, "-", "+"]]) + "" + (LaTeX[abs(b)]) + "z=1  \\ " + (LaTeX[c]) + "x" + (LaTeX[If[d < 0, "-", "+"]]) + "" + (LaTeX[abs(d)]) + "z=0 \\ \end{array}   \right.\ \;,\;\left\lbrace \begin{array}{ll}" + (LaTeX[a]) + " y" + (LaTeX[If[b < 0, "-", "+"]]) + "" + (LaTeX[abs(b)]) + "u =0 \\ " + (LaTeX[c]) + "y" + (LaTeX[If[d < 0, "-", "+"]]) + "" + (LaTeX[abs(d)]) + "u =1 \\ \end{array}  \right.\  $"
T9: "$ x=\frac{" + (LaTeX[d]) + "}{" + (LaTeX[det]) + "} =" + (LaTeX[FractionText[d / det]]) + "\;,\;y=\frac{" + (LaTeX[-b]) + "}{" + (LaTeX[det]) + "}=" + (LaTeX[FractionText[(-b) / det]]) + "\;,\;z=\frac{" + (LaTeX[-c]) + "}{" + (LaTeX[det]) + "}=" + (LaTeX[FractionText[(-c) / det]]) + "\;,\;u=\frac{" + (LaTeX[a]) + "}{" + (LaTeX[det]) + "}=" + (LaTeX[FractionText[a / det]]) + "$"
e = false
f = true
T6: "$ \Delta=\left\vert\begin{array}{cc} " + (LaTeX[a]) + " & " + (LaTeX[b]) + " \\ " + (LaTeX[c]) + " & " + (LaTeX[d]) + " \\ \end{array}   \right\vert =" + (LaTeX[det]) + " \;$"
T7: "$ \Delta_x=\left\vert\begin{array}{cc}1 & " + (LaTeX[b]) + " \\0 &" + (LaTeX[d]) + " \\ \end{array}   \right\vert =" + (LaTeX[d]) + " \;,\;\Delta_y=\left\vert \begin{array}{cc} 0 & " + (LaTeX[b]) + " \\ 1 & " + (LaTeX[d]) + "\\ \end{array}  \right\vert=" + (LaTeX[-b]) + " $"
T8: "$ \Delta_z=\left\vert  \begin{array}{cc} " + (LaTeX[a]) + " & 1 \\ " + (LaTeX[c]) + " & 0 \\ \end{array} \right\vert =" + (LaTeX[-c]) + "\;,\;\Delta_u=\left\vert \begin{array}{cc} " + (LaTeX[a]) + " & 0 \\ " + (LaTeX[c]) + " & 1 \\ \end{array}  \right\vert =" + (LaTeX[a]) + " $"
m = 0 (slider from 0 to If[det ≟ 0, 6, 10], step 1)
T10: "$ ⇒A^{-1}=\left [ \begin{array}{cc} " + (LaTeX[FractionText[d / det]]) + " & " + (LaTeX[FractionText[(-b) / det]]) + " \\ " + (LaTeX[FractionText[(-c) / det]]) + " & " + (LaTeX[FractionText[a / det]]) + " \\ \end{array}  \right ]  $"
text3: "" + (LaTeX[If[det ≟ 0, "反方陣不存在", ""]]) + ""
T1: "$ ⇒A=\left [  \begin{array}{cc} " + (LaTeX[a]) + " & " + (LaTeX[b]) + " \\ " + (LaTeX[c]) + " & " + (LaTeX[d]) + " \\ \end{array} \right ] ,\;設A^{-1}=\left [ \begin{array}{cc} x& y \\ z & u \\ \end{array}  \right ]  $"