三階行列式的展開
Author: 藍邦偉  老師

三階行列式的性質(拉動s重新整理)

1.行列互換，其值不變。

2.某兩行（列）對調，其值變號。

3. 任一行(列)可以提出公因數。

k=

4. 對任一行或任一列都有分配律。

5. 將任一行(列)乘上k倍加到另一行(列)，其值不變。

s = 0 (slider from 0 to 1, step 0.01)
a_2: Random[-5, 5] + s * 0
a_3: Random[-5, 5] + 0 * s
b_1 = Random(-5, 5)
b_2: Random[-5, 5] + s * 0
b_3: Random[-5, 5] + s * 0
c_1: Random[-5, 5] + s * 0
a_1: Random[-5, 5] + s * 0
c_2 = Random(-5, 5)
c_3: Random[-5, 5] + s * 0
text1: "\left|\begin{array}{rr}" + a_1 + " &" + a_2 + "  &" + a_3 + "  \\ " + b_1 + "  &" + b_2 + "  &" + b_3 + "  \\ " + c_1 + "&" + c_2 + "  & " + c_3 + " \\ \end{array}\right|"
n = 0 (slider from 0 to 5, step 1)
d1: a_1 * b_2 * c_3
d2: a_3 * b_1 * c_2
d3: a_2 * b_3 * c_1
d4: a_3 * b_2 * c_1
d5: a_2 * b_1 * c_3
d6: a_1 * b_3 * c_2
text3: "$ \left\vert \begin{array}{rr}" + (LaTeX[a_1]) + " &" + (LaTeX[b_1]) + " &" + (LaTeX[c_1]) + " \\" + (LaTeX[a_2]) + " & " + (LaTeX[b_2]) + " &" + (LaTeX[c_2]) + " \\ " + (LaTeX[a_3]) + " &" + (LaTeX[b_3]) + " &" + (LaTeX[c_3]) + "\\ \end{array} \right\vert  $"
文字4: "= " + (LaTeX[a_1 * b_2 * c_3 + a_2 * b_3 * c_1 + a_3 * b_1 * c_2 - a_3 * b_2 * c_1 - a_2 * b_1 * c_3 - a_1 * b_3 * c_2]) + ""
text4: "=" + (LaTeX[d1]) + "" + (LaTeX[If[d2 < 0, "-", "+"]]) + "" + (LaTeX[abs(d2)]) + "" + (LaTeX[If[d3 < 0, "-", "+"]]) + "" + (LaTeX[abs(d3)]) + "" + (LaTeX[If[d4 < 0, "+", "-"]]) + "" + (LaTeX[abs(d4)]) + "" + (LaTeX[If[d5 < 0, "+", "-"]]) + "" + (LaTeX[abs(d5)]) + "" + (LaTeX[If[d6 < 0, "+", "-"]]) + "" + (LaTeX[abs(d6)]) + ""
text5: "=" + (LaTeX[d1 + d2 + d3 - d4 - d5 - d6]) + ""
text7: "\left\vert \begin{array}{rr}" + a_1 + " &" + a_3 + " &" + a_2 + "\\ " + b_1 + "&" + b_3 + "&" + b_2 + " \\ " + c_1 + "& " + c_3 + "&" + c_2 + "\\ \end{array} \right\vert "
text9: "\left\vert \begin{array}{rr}" + c_1 + " &" + c_2 + " &" + c_3 + " \\ " + b_1 + " &" + b_2 + " &" + b_3 + " \\" + a_1 + " &" + a_2 + " &" + a_3 + " \\ \end{array} \right\vert "
text8: "=" + (LaTeX[d6]) + "" + (LaTeX[If[d4 < 0, "-", "+"]]) + "" + (LaTeX[abs(d4)]) + "" + (LaTeX[If[d5 < 0, "-", "+"]]) + "" + (LaTeX[abs(d5)]) + "" + (LaTeX[If[d3 > 0, "-", "+"]]) + "" + (LaTeX[abs(d3)]) + "" + (LaTeX[If[d2 > 0, "-", "+"]]) + "" + (LaTeX[abs(d2)]) + "" + (LaTeX[If[d1 > 0, "-", "+"]]) + "" + (LaTeX[abs(d1)]) + "=" + (LaTeX[d5 + d4 + d6 - d3 - d2 - d1]) + ""
text10: "=" + d4 + (If[d6 < 0, "-", "+"]) + (abs(d6)) + (If[d5 < 0, "-", "+"]) + (abs(d5)) + (If[d1 > 0, "-", "+"]) + (abs(d1)) + (If[d3 > 0, "-", "+"]) + (abs(d3)) + (If[d2 > 0, "-", "+"]) + (abs(d2)) + "=" + (d4 + d6 + d5 - d1 - d2 - d3)
k = 1
text12: "\left\vert\begin{array}{rr}" + a_1 + " &" + a_2 + " &" + a_3 + "\\" + (k * b_1) + " &" + (k * b_2) + " &" + (k * b_3) + " \\ " + c_1 + " &" + c_2 + "&" + c_3 + " \\ \end{array}  \right\vert "
inputBox1 = Textfield(k)
text13: "=" + (k * d1) + (If[d3 < 0, "-", "+"]) + (abs(k * d3)) + (If[d2 < 0, "-", "+"]) + (abs(k * d2)) + (If[d4 > 0, "+", "-"]) + (abs(k * d4)) + (If[d6 < 0, "+", "-"]) + (abs(k * d6)) + (If[d5 < 0, "+", "-"]) + (abs(k * d5)) + "=" + (k * d1 + k * d2 + k * d3 - k * d4 - k * d5 - k * d6)
text14: "\left\vert\begin{array}{rr}" + (LaTeX[k * a_1]) + " &" + (LaTeX[a_2]) + " &" + (LaTeX[a_3]) + " \\" + (LaTeX[k * b_1]) + "&" + (LaTeX[b_2]) + " &" + (LaTeX[b_3]) + " \\" + (LaTeX[k * c_1]) + "&" + (LaTeX[c_2]) + " &" + (LaTeX[c_3]) + " \\ \end{array}  \right\vert "
text15: "=" + (LaTeX[k * d1]) + "" + (LaTeX[If[d3 < 0, "-", "+"]]) + "" + (LaTeX[abs(k * d3)]) + "" + (LaTeX[If[d2 < 0, "-", "+"]]) + "" + (LaTeX[abs(k * d2)]) + "" + (LaTeX[If[d4 > 0, "-", "+"]]) + "" + (LaTeX[abs(k * d4)]) + "" + (LaTeX[If[d5 > 0, "-", "+"]]) + "" + (LaTeX[abs(k * d5)]) + "" + (LaTeX[If[d6 > 0, "-", "+"]]) + "" + (LaTeX[abs(k * d6)]) + "=" + (LaTeX[k * (d1 + d2 + d3 - d4 - d5 - d6)]) + ""
e: Random[1, 5] + s * 0
f: Random[1, 5] + s * 0
g: Random[1, 5] + s * 0
text17: "\left\vert  \begin{array}{rr}" + (LaTeX[a_1]) + "+ " + (LaTeX[e]) + " &" + (LaTeX[a_2]) + " &" + (LaTeX[a_3]) + " \\ " + (LaTeX[b_1]) + "+ " + (LaTeX[f]) + "&" + (LaTeX[b_2]) + "&" + (LaTeX[b_3]) + " \\" + (LaTeX[c_1]) + "+" + (LaTeX[g]) + "&" + (LaTeX[c_2]) + " & " + (LaTeX[c_3]) + " \\ \end{array}\right\vert "
text18: "\left\vert\begin{array}{rr}" + a_1 + " &" + a_2 + "&" + a_3 + " \\" + b_1 + " &" + b_2 + " &" + b_3 + " \\ " + c_1 + " &" + c_2 + "&" + c_3 + " \\ \end{array}  \right\vert "
text19: "+" + "\left\vert \begin{array}{rr}" + e + " &" + a_2 + " &" + a_3 + " \\" + f + "&" + b_2 + " &" + b_3 + " \\" + g + " &" + c_2 + " &" + c_3 + " \\ \end{array} \right\vert "
j1: e * b_2 * c_3
j2: b_3 * a_2 * g
j3: a_3 * f * c_2
j4: a_3 * b_2 * g
j5: b_3 * c_2 * e
j6: c_3 * a_2 * f
text21: "+(" + (LaTeX[j1]) + "" + (LaTeX[If[j2 < 0, "-", "+"]]) + "" + (LaTeX[abs(j2)]) + "" + (LaTeX[If[j3 < 0, "-", "+"]]) + "" + (LaTeX[abs(j3)]) + "" + (LaTeX[If[j4 > 0, "-", "+"]]) + "" + (LaTeX[abs(j4)]) + "" + (LaTeX[If[j5 > 0, "-", "+"]]) + "" + (LaTeX[abs(j5)]) + "" + (LaTeX[If[j6 > 0, "-", "+"]]) + "" + (LaTeX[abs(j6)]) + ")"
text20: "=(" + (LaTeX[d1]) + "" + (LaTeX[If[d3 < 0, "-", "+"]]) + "" + (LaTeX[abs(d3)]) + "" + (LaTeX[If[d2 < 0, "-", "+"]]) + "" + (LaTeX[abs(d2)]) + "" + (LaTeX[If[d4 > 0, "-", "+"]]) + "" + (LaTeX[abs(d4)]) + "" + (LaTeX[If[d5 > 0, "-", "+"]]) + "" + (LaTeX[abs(d5)]) + "" + (LaTeX[If[d6 > 0, "-", "+"]]) + "" + (LaTeX[abs(d6)]) + ")"
text22: "=" + (LaTeX[d1 + d2 + d3 - d4 - d5 - d6]) + "" + (LaTeX[If[j1 + j2 + j3 - j4 - j5 - j6 < 0, "-", "+"]]) + "" + (LaTeX[abs(j1 + j2 + j3 - j4 - j5 - j6)]) + ""
text23: "=" + (LaTeX[d1 + d2 + d3 - d4 - d5 - d6 + j1 + j2 + j3 - j4 - j5 - j6]) + ""
k1: (a_1 + e) * b_2 * c_3
k2: a_2 * b_3 * (g + c_1)
k3: a_3 * (f + b_1) * c_2
k4: a_3 * b_2 * (g + c_1)
k5: b_3 * c_2 * (e + a_1)
k6: c_3 * a_2 * (f + b_1)
text24: "=" + (LaTeX[k1]) + "" + (LaTeX[If[k2 < 0, "-", "+"]]) + "" + (LaTeX[abs(k2)]) + "" + (LaTeX[If[k3 < 0, "-", "+"]]) + "" + (LaTeX[abs(k3)]) + "" + (LaTeX[If[k4 > 0, "-", "+"]]) + "" + (LaTeX[abs(k4)]) + "" + (LaTeX[If[k5 < 0, "-", "+"]]) + "" + (LaTeX[abs(k5)]) + "" + (LaTeX[If[k6 < 0, "-", "+"]]) + "" + (LaTeX[abs(k6)]) + ""
text25: "=" + (k1 + k2 + k3 - k4 - k5 - k6)
文字3: "= " + (LaTeX[a_1 * b_2 * c_3]) + "" + (LaTeX[If[a_2 * b_3 * c_1 < 0, "-", "+"]]) + "" + (LaTeX[abs(a_2 * b_3 * c_1)]) + "" + (LaTeX[If[a_3 * b_1 * c_2 < 0, "-", "+"]]) + "" + (LaTeX[abs(a_3 * b_1 * c_2)]) + "" + (LaTeX[If[a_3 * b_2 * c_1 < 0, "+", "-"]]) + "" + (LaTeX[abs(a_3 * b_2 * c_1)]) + "" + (LaTeX[If[a_2 * b_1 * c_3 < 0, "+", "-"]]) + "" + (LaTeX[abs(a_2 * b_1 * c_3)]) + "" + (LaTeX[If[a_1 * b_3 * c_2 < 0, "+", "-"]]) + "" + (LaTeX[abs(a_1 * b_3 * c_2)]) + ""
u1: a_1 * (b_2 + k * b_3) * c_3
u2: (a_2 + k * a_3) * b_3 * c_1
u3: a_3 * b_1 * (c_2 + k * c_3)
u4: a_3 * (b_2 + k * b_3) * c_1
u5: b_3 * (c_2 + k * c_3) * a_1
u6: c_3 * (a_2 + k * a_3) * b_1
text27: " \left\vert\begin{array}{rr}" + (LaTeX[a_1]) + " &" + (LaTeX[a_2]) + "" + (LaTeX[If[k * a_3 < 0, "-", "+"]]) + "" + (LaTeX[abs(k)]) + "×" + (LaTeX[abs(a_3)]) + "&" + (LaTeX[a_3]) + " \\" + (LaTeX[b_1]) + "&" + (LaTeX[b_2]) + "" + (LaTeX[If[k * b_2 < 0, "-", "+"]]) + "" + (LaTeX[abs(k)]) + "×" + (LaTeX[abs(b_3)]) + " &" + (LaTeX[b_3]) + " \\" + (LaTeX[c_1]) + "&" + (LaTeX[c_2]) + "" + (LaTeX[If[k * c_3 < 0, "-", "+"]]) + "" + (LaTeX[abs(k)]) + "×" + (LaTeX[abs(c_3)]) + " &" + (LaTeX[c_3]) + " \\ \end{array}  \right\vert "
text28: "=" + (LaTeX[u1]) + "" + (LaTeX[If[u2 < 0, "-", "+"]]) + "" + (LaTeX[abs(u2)]) + "" + (LaTeX[If[u3 < 0, "-", "+"]]) + "" + (LaTeX[abs(u3)]) + "" + (LaTeX[If[u4 > 0, "-", "+"]]) + "" + (LaTeX[abs(u4)]) + "" + (LaTeX[If[u5 < 0, "+", "-"]]) + "" + (LaTeX[abs(u5)]) + "" + (LaTeX[If[u6 < 0, "+", "-"]]) + "" + (LaTeX[abs(u6)]) + ""
text29: "=" + (LaTeX[u1 + u2 + u3 - u4 - u5 - u6]) + ""