二階行列式的性質
Author: 藍邦偉  老師

性質(1) 行列互換，其值不變

性質(2) 兩行或兩列互換，其值變號

性質(3) 任一行列可以提出同一數

性質(4) 任兩行列成比例，其值為0

性質(5) 任一行(列)乘上同一數加到另一行列，其值不變

\left|\begin{array}{rr} a & b \ c & d \ \end{array}\right|=\left|\begin{array}{rr} a & c \ b & d \ \end{array}\right|

\left|\begin{array}{rr} a & b \ c & d \ \end{array}\right|=-\left|\begin{array}{rr} c & d \ a & b \ \end{array}\right|

\left|\begin{array}{rr} a & b \ c & d \ \end{array}\right|=-\left|\begin{array}{rr} b & a \ d & c \ \end{array}\right|

\left|\begin{array}{rr} a & kb \ c & kd \ \end{array}\right|=k \left|\begin{array}{rr} a & b \ c & d \ \end{array}\right|

\left|\begin{array}{rr} ka & kb \ c & d \ \end{array}\right|=k \left|\begin{array}{rr} a & b \ c & d \ \end{array}\right|

\left|\begin{array}{rr} a & b \ ka & kb \ \end{array}\right|=kab-kab=0

\left|\begin{array}{rr}a & ka \ b & kb \ \end{array}\right|=kab-kab=0

\left|\begin{array}{rr} a & b \ c & d \ \end{array}\right|=ad-bc=\left|\begin{array}{rr} a & b+ka \ c & d+kc \ \end{array}\right|=a(d+kc)-(b+ka)c

\left|\begin{array}{rr} a & b \ c & d \ \end{array}\right|=ad-bc=\left|\begin{array}{rr} a & b \ ka+c & kb+d \ \end{array}\right|=a(kb+d)-b(ka+c)

二階行列式的性質

s = 0.5 (slider from 0 to 1, step 0.01)
n = 0 (slider from 0 to 5, step 1)
a: Random[-5, 5] + s * 0
c: Random[-5, 5] + 0
b: Random[-5, 5] + s * 0
d: Random[0, 5] + s * 0
h = Random(-5, 5)
k = Random(-5, 5)
文字1: "\left|\begin{array}{rr} " + a + " & " + b + " \\" + c + " & " + d + " \\ \end{array}\right|=\left|\begin{array}{rr} " + a + " &" + c + " \\" + b + " &" + d + "  \\ \end{array}\right|=(" + a + ")(" + d + ")-(" + b + ")(" + c + ")=" + (a * d) + (If[b * c > 0, "-", "+"]) + (abs(b * c)) + "=" + (a * d - b * c)
文字3: "\left|\begin{array}{rr}" + c + " &" + d + " \\" + a + " &" + b + " \\ \end{array}\right|=" + (c * d) + (If[a * b > 0, "-", "+"]) + (abs(a * b)) + "=" + (c * b - a * d)
文字6: "\left|\begin{array}{rr}" + b + " &" + a + " \\" + d + "&" + c + " \\ \end{array}\right|=" + (c * b) + (If[a * d > 0, "-", "+"]) + (abs(a * d)) + "=" + (b * c - a * d)
文字9: "\left|\begin{array}{rr}" + a + " &" + (h * b) + " \\" + c + " &" + (h * d) + " \\ \end{array}\right|=" + h + "\left|\begin{array}{rr}" + a + "&" + b + " \\" + c + "&" + d + "\\ \end{array}\right|=" + h + "(" + (a * d) + (If[b * c > 0, "-", "+"]) + (abs(b * c)) + ")=" + (h * (a * d - b * c))
文字12: "\left|\begin{array}{rr}" + (h * a) + "&" + (h * b) + "\\" + c + " &" + d + " \\ \end{array}\right|=" + h + "\left|\begin{array}{rr} " + a + " &" + b + " \\" + c + " &" + d + " \\ \end{array}\right|=" + h + "(" + (a * d) + (If[b * c > 0, "-", "+"]) + (abs(b * c)) + ")=" + (h * (a * d - b * c))
文字13: "\left|\begin{array}{rr}" + a + "&" + (h * a) + " \\" + c + " &" + (h * c) + "\\ \end{array}\right|=" + (h * a * c) + (If[h * a * c < 0, "+", "-"]) + (abs(h * a * c)) + "=0"
文字14: "\left|\begin{array}{rr}" + a + " &" + b + " \\" + (k * a) + " &" + (k * b) + " \\ \end{array}\right|=" + (k * a * b) + (If[k * a * b < 0, "+", "-"]) + (abs(k * a * b)) + "=0"
文字15: "\left|\begin{array}{rr}" + a + " &" + b + " \\" + c + " &" + d + " \\ \end{array}\right|=" + (a * d - b * c) + ",\left|\begin{array}{rr} " + a + " &" + b + " \\" + c + " &" + d + " \\ \end{array}\right|=\left|\begin{array}{rr}" + a + " &" + b + "+(" + (h * a) + " )\\" + c + " &" + d + "+(" + (h * c) + ") \\ \end{array}\right|=\left|\begin{array}{rr}" + a + " &" + (b + h * a) + "\\" + c + " &" + (d + h * c) + " \\ \end{array}\right|=" + (a * (d + h * c)) + (If[c * (b + h * a) > 0, "-", "+"]) + (abs(c * (b + h * a))) + "=" + (a * d - b * c)
文字16: "\left|\begin{array}{rr} " + a + " &" + b + " \\" + c + " &" + d + "\\ \end{array}\right|=" + (a * d - b * c) + ",\left|\begin{array}{rr} " + a + " &" + b + " \\" + c + " &" + d + " \\ \end{array}\right|=\left|\begin{array}{c} " + a + "+(" + (k * c) + ")&" + b + "+(" + (k * d) + ")\\" + c + " &" + d + " \\ \end{array}\right|=\left|\begin{array}{rr}" + (a + k * c) + " &" + (b + k * d) + " \\" + c + " &" + d + " \\ \end{array}\right|=" + (d * (k * c + a)) + (If[b * k * d + c > 0, "+", "-"]) + (abs(c * (k * d + b))) + "=" + (a * d - b * c)
文字4: "\left|\begin{array}{rr}" + (LaTeX[a]) + " & " + (LaTeX[b]) + " \\" + (LaTeX[c]) + " &" + (LaTeX[d]) + " \\ \end{array}\right|=" + (LaTeX[a * d]) + "" + (LaTeX[If[b * c > 0, "-", "+"]]) + "" + (LaTeX[abs(b * c)]) + "=" + (LaTeX[a * d - b * c]) + ""