行列式とは？意味と定義・求め方・性質を徹底解説

\[\begin{eqnarray}
\left[ \begin{array}{cc} 1 & 3 \\ -2 & 1 \end{array} \right]
\end{eqnarray}\]

\[\begin{eqnarray}
\left| \begin{array}{cc} 1 & 3 \\ -2 & 1 \end{array} \right|
&=& 1\cdot1-3\cdot(-2)\\
&=& 1+6\\
&=&7
\end{eqnarray}\]

\[\begin{eqnarray}
\left[ \begin{array}{cc} 2 & 0 \\ -2 & -1 \end{array} \right]
\end{eqnarray}\]

\[\begin{eqnarray}
\left| \begin{array}{cc} 2 & 0 \\ -2 & -1 \end{array} \right|
&=& 2\cdot(-1)-0\cdot(-2)\\
&=& -2+0\\
&=&-2
\end{eqnarray}\]

\[\begin{eqnarray}
\left[ \begin{array}{cc} 1 & 2 \\ 1 & 2 \end{array} \right]
\end{eqnarray}\]

\[\begin{eqnarray}
\left| \begin{array}{cc} 1 & 2 \\ 1 & 2 \end{array} \right|
&=& 1\cdot2-2\cdot1\\
&=& 2-2\\
&=&-0
\end{eqnarray}\]

\[\begin{eqnarray}
\left[ \begin{array}{cc}
1 & -3 & 1\\
-2 & 1 & 0 \\
2 & 1 & -2
\end{array} \right]
\end{eqnarray}\]

\[\begin{eqnarray}
\left| \begin{array}{cc}
1 & -3 & 1\\
-2 & 1 & 0 \\
2 & 1 & -2
\end{array} \right|
&=&
(1\cdot1\cdot-2)+ (-3\cdot0\cdot2) + (1\cdot-2\cdot1)
– (-3\cdot-2\cdot-2) – (1\cdot0\cdot1)-(1\cdot1\cdot2)\\
&=& -2+0-2+12-0-2\\
&=&6
\end{eqnarray}\]

\[\begin{eqnarray}
\left[ \begin{array}{cc}
1 & 3 & 1\\
2 & 1 & 0 \\
1 & 1 & 2
\end{array} \right]
\end{eqnarray}\]

\[\begin{eqnarray}
\left| \begin{array}{cc}
1 & 3 & 1\\
2 & 1 & 0 \\
1 & 1 & 2
\end{array} \right|
&=&
(1\cdot1\cdot2 )+ (3\cdot0\cdot1) + (1\cdot2\cdot1)
– (3\cdot2\cdot2) – (1\cdot0\cdot1)-(1\cdot1\cdot1)\\
&=& 2+0+2-12-0-1\\
&=&-9
\end{eqnarray}\]

\[\begin{eqnarray}
\left[ \begin{array}{cc}
1 & 3 & 2\\
2 & 1 & 4 \\
1 & 1 & 2
\end{array} \right]
\end{eqnarray}\]

\[\begin{eqnarray}
\left| \begin{array}{cc}
1 & 3 & 2\\
2 & 1 & 4 \\
1 & 1 & 2
\end{array} \right|
&=&
(1\cdot1\cdot2 )+ (3\cdot4\cdot1) + (2\cdot2\cdot1)
– (3\cdot2\cdot2) – (1\cdot4\cdot1)-(2\cdot1\cdot1)\\
&=& 2+12+4-12-4-2\\
&=&-0
\end{eqnarray}\]

\[\begin{eqnarray}
A&=&\left[ \begin{array}{cc} a & b \\ c & d \end{array} \right],
\hspace{4mm}
|A|=ad-bc \\
A^T&=&\left[ \begin{array}{cc} a & c \\ b & d \end{array} \right],
\hspace{4mm}
|A^T|=ad-cb
\end{eqnarray}\]

\[\begin{eqnarray}
A&=&\left[ \begin{array}{cc}
a & b & c \\
d & e & f \\
g & h & i
\end{array} \right],
\hspace{4mm}
|A|=a(ei-hf)-b(di-gf)+c(dh-ge)\\
A^T&=&\left[ \begin{array}{cc}
a & d & g \\
b & e & h \\
c & f & i
\end{array} \right],
\hspace{4mm}
|A^T|=a(ei-fh)-b(di-fg)+c(dh-eg)\\
\end{eqnarray}\]

\[\begin{eqnarray}
AB
&=&
\left[ \begin{array}{cc} a & b \\ c & d \end{array} \right]
\left[ \begin{array}{cc} e & f \\ g & h \end{array} \right]
=
\left[ \begin{array}{cc} ae+bg & af+bh \\ ce+dg & cf+dh \end{array} \right]
\end{eqnarray}\]

\[\begin{eqnarray}
|AB|
&=&
\left| \begin{array}{cc} ae+bg & af+bh \\ ce+dg & cf+dh \end{array} \right|\\
&=&
(ae+bg)(cf+dh)-(af+bh)(ce+dg)\\
&=&
adeh-adfg-bceh+bcfg
\end{eqnarray}\]

\[\begin{eqnarray}
|A|
&=&
\left| \begin{array}{cc} a & b \\ c & d \end{array} \right|
=ad-bc\\
|B|
&=&
\left| \begin{array}{cc} e &f \\ g & h \end{array} \right|
=eh-fg
\end{eqnarray}\]

\[\begin{eqnarray}
|A||B|
&=&
(ad-bc)(eh-fg)\\
&=&
adeh-adfg-bceh+bcfg
\end{eqnarray}\]

\[\begin{eqnarray}
AB
&=&
\left[ \begin{array}{cc}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array} \right]
\left[ \begin{array}{cc}
b_{11} & b_{12} & b_{13} \\
b_{21} & b_{22} & b_{23} \\
b_{31} & b_{32} & b_{33}
\end{array} \right]\\
&=&
\left[ \begin{array}{cc}
a_{11}b_{11}+a_{12}b_{21}+a_{13}b_{31}
& a_{11}b_{12}+a_{12}b_{22}+a_{13}b_{32}
& a_{11}b_{13}+a_{12}b_{23}+a_{13}b_{33} \\
a_{21}b_{11}+a_{22}b_{21}+a_{23}b_{32}
& a_{21}b_{12}+a_{22}b_{22}+a_{23}b_{32}
& a_{21}b_{13}+a_{22}b_{23}+a_{23}b_{33} \\
+a_{31}b_{11}+a_{32}b_{21}+a_{33}b_{31}
&a_{31}b_{12}+a_{32}b_{22}+a_{33}b_{32}
&a_{31}b_{13}+a_{32}b_{23}+a_{33}b_{33}
\end{array} \right]

\end{eqnarray}\]

\[|AB|=\\
a_{11} a_{22} a_{33} b_{11} b_{22} b_{33} – a_{11} a_{22} a_{33} b_{11} b_{23} b_{32}-a_{11} a_{22} a_{33} b_{12} b_{21} b_{33} + a_{11} a_{22} a_{33} b_{12} b_{23} b_{31}\\
+ a_{11} a_{22} a_{33} b_{13} b_{21} b_{32} – a_{11} a_{22} a_{33} b_{13} b_{22} b_{31} – a_{11} a_{23} a_{32} b_{11} b_{22} b_{33} + a_{11} a_{23} a_{32} b_{11} b_{23} b_{32}\\
+ a_{11} a_{23} a_{32} b_{12} b_{21} b_{33} – a_{11} a_{23} a_{32} b_{12} b_{23} b_{31} – a_{11} a_{23} a_{32} b_{13} b_{21} b_{32} + a_{11} a_{23} a_{32} b_{13} b_{22} b_{31}\\
– a_{12} a_{21} a_{33} b_{11} b_{22} b_{33} + a_{12} a_{21} a_{33} b_{11} b_{23} b_{32} + a_{12} a_{21} a_{33} b_{12} b_{21} b_{33} – a_{12} a_{21} a_{33} b_{12} b_{23} b_{31}\\
– a_{12} a_{21} a_{33} b_{13} b_{21} b_{32} + a_{12} a_{21} a_{33} b_{13} b_{22} b_{31} + a_{12} a_{23} a_{31} b_{11} b_{22} b_{33} – a_{12} a_{23} a_{31} b_{11} b_{23} b_{32}\\
– a_{12} a_{23} a_{31} b_{12} b_{21} b_{33} + a_{12} a_{23} a_{31} b_{12} b_{23} b_{31} + a_{12} a_{23} a_{31} b_{13} b_{21} b_{32} – a_{12} a_{23} a_{31} b_{13} b_{22} b_{31}\\
+ a_{13} a_{21} a_{32} b_{11} b_{22} b_{33} – a_{13} a_{21} a_{32} b_{11} b_{23} b_{32} – a_{13} a_{21} a_{32} b_{12} b_{21} b_{33} + a_{13} a_{21} a_{32} b_{12} b_{23} b_{31}\\
+ a_{13} a_{21} a_{32} b_{13} b_{21} b_{32} – a_{13} a_{21} a_{32} b_{13} b_{22} b_{31} – a_{13} a_{22} a_{31} b_{11} b_{22} b_{33} + a_{13} a_{22} a_{31} b_{11} b_{23} b_{32}\\
+ a_{13} a_{22} a_{31} b_{12} b_{21} b_{33} – a_{13} a_{22} a_{31} b_{12} b_{23} b_{31} – a_{13} a_{22} a_{31} b_{13} b_{21} b_{32} + a_{13} a_{22} a_{31} b_{13} b_{22} b_{31}
\]

\[\begin{eqnarray}
|A|
&=&
\left[ \begin{array}{cc}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array} \right]
=a_{11}a_{22}a_{33}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{13}a_{22}a_{31}\\
|B|
&=&
\left[ \begin{array}{cc}
b_{11} & b_{12} & b_{13} \\
b_{21} & b_{22} & b_{23} \\
b_{31} & b_{32} & b_{33}
\end{array} \right]
=
b_{11}b_{22}b_{33}-b_{11}b_{23}b_{32}-b_{12}b_{21}b_{33}+b_{12}b_{23}b_{31}+b_{13}b_{21}b_{32}-b_{13}b_{22}b_{31}\\
\end{eqnarray}\]

\[|A||B|=\\
a_{11} a_{22} a_{33} b_{11} b_{22} b_{33} – a_{11} a_{22} a_{33} b_{11} b_{23} b_{32}-a_{11} a_{22} a_{33} b_{12} b_{21} b_{33} + a_{11} a_{22} a_{33} b_{12} b_{23} b_{31}\\
+ a_{11} a_{22} a_{33} b_{13} b_{21} b_{32} – a_{11} a_{22} a_{33} b_{13} b_{22} b_{31} – a_{11} a_{23} a_{32} b_{11} b_{22} b_{33} + a_{11} a_{23} a_{32} b_{11} b_{23} b_{32}\\
+ a_{11} a_{23} a_{32} b_{12} b_{21} b_{33} – a_{11} a_{23} a_{32} b_{12} b_{23} b_{31} – a_{11} a_{23} a_{32} b_{13} b_{21} b_{32} + a_{11} a_{23} a_{32} b_{13} b_{22} b_{31}\\
– a_{12} a_{21} a_{33} b_{11} b_{22} b_{33} + a_{12} a_{21} a_{33} b_{11} b_{23} b_{32} + a_{12} a_{21} a_{33} b_{12} b_{21} b_{33} – a_{12} a_{21} a_{33} b_{12} b_{23} b_{31}\\
– a_{12} a_{21} a_{33} b_{13} b_{21} b_{32} + a_{12} a_{21} a_{33} b_{13} b_{22} b_{31} + a_{12} a_{23} a_{31} b_{11} b_{22} b_{33} – a_{12} a_{23} a_{31} b_{11} b_{23} b_{32}\\
– a_{12} a_{23} a_{31} b_{12} b_{21} b_{33} + a_{12} a_{23} a_{31} b_{12} b_{23} b_{31} + a_{12} a_{23} a_{31} b_{13} b_{21} b_{32} – a_{12} a_{23} a_{31} b_{13} b_{22} b_{31}\\
+ a_{13} a_{21} a_{32} b_{11} b_{22} b_{33} – a_{13} a_{21} a_{32} b_{11} b_{23} b_{32} – a_{13} a_{21} a_{32} b_{12} b_{21} b_{33} + a_{13} a_{21} a_{32} b_{12} b_{23} b_{31}\\
+ a_{13} a_{21} a_{32} b_{13} b_{21} b_{32} – a_{13} a_{21} a_{32} b_{13} b_{22} b_{31} – a_{13} a_{22} a_{31} b_{11} b_{22} b_{33} + a_{13} a_{22} a_{31} b_{11} b_{23} b_{32}\\
+ a_{13} a_{22} a_{31} b_{12} b_{21} b_{33} – a_{13} a_{22} a_{31} b_{12} b_{23} b_{31} – a_{13} a_{22} a_{31} b_{13} b_{21} b_{32} + a_{13} a_{22} a_{31} b_{13} b_{22} b_{31}
\]

\[\begin{eqnarray}
|A|
&=&
\left| \begin{array}{cc} a_1 + a_2 & b_1+b_2 \\ c & d \end{array} \right|
=
(a_1+a_2)d-(b_1+b_2)c\\
&\rightarrow&
\left| \begin{array}{cc} a_1 & b_1 \\ c & d \end{array} \right|
+
\left| \begin{array}{cc} a_2 & b_2 \\ c & d \end{array} \right|
=
(a_1d-b_1c)+(a_2d-b_2c)\\
\end{eqnarray}\]

\[\begin{eqnarray}
|A|
&=&
\left| \begin{array}{cc} a & b_1+b_2 \\ c & d_1+d_2 \end{array} \right|
=
a(d_1+d_2)-(b_1+b_2)c\\
&\rightarrow&
\left| \begin{array}{cc} a & b_1 \\ c & d_1 \end{array} \right|
+
\left| \begin{array}{cc} a & b_2 \\ c & d_2 \end{array} \right|
=
(ad_1-b_1c)+(ad_2-b_2c)\\
\end{eqnarray}\]

\[\begin{eqnarray}
|A|
&=&
\left| \begin{array}{cc}
a_1+a_2 & b_1+b_2 & c_1+c_2 \\
d & e & f \\
g & h & i
\end{array} \right|
=
(a_1+a_2)(ei-fh)-(b_1+b_2)(di-fg)+(c_1+c_2)(dh-eg)\\
&\rightarrow&
\left| \begin{array}{cc}
a_1 & b_1 & c_1 \\
d & e & f \\
g & h & i
\end{array} \right|
+
\left| \begin{array}{cc}
a_2 & b_2 & c_2 \\
d & e & f \\
g & h & i
\end{array} \right|
=(a_1(ei-fh)-b_1(di-fg)+c_1(dh-eg))+(a_2(ei-fh)-b_2(di-fg)+c_2(dh-eg))\\
\end{eqnarray}\]

\[\begin{eqnarray}
|A|
&=&
\left| \begin{array}{cc}
a & b & c_1+c_2 \\
d & e & f_1+f_2\\
g & h & i_1+i_2
\end{array} \right|
=
(c_1+c_2)(dh-eg)-(f_1+f_2)(ah-bg)+(i_1+i_2)(ae-bd)\\
&\rightarrow&
\left| \begin{array}{cc}
a & b & c_1 \\
d & e & f_1 \\
g & h & i_1
\end{array} \right|
+
\left| \begin{array}{cc}
a & b & c_2 \\
d & e & f_2 \\
g & h & i_2
\end{array} \right|
=(c_1(dh-eg)-f_1(ah-bg)+i_1(ae-bd))+(c_2(dh-eg)-f_2(ah-bg)+i_2(ae-bd))\\
\end{eqnarray}\]

\[\begin{eqnarray}
|A|
&=&
\left| \begin{array}{cc} a & b \\ c & d \end{array} \right|
= ad-bc \\
|A^1|
&=&
\left| \begin{array}{cc} c & d \\ a & b \end{array} \right|
= cb-da = -(ad-bc)
\end{eqnarray}\]

\[\begin{eqnarray}
|A|
&=&
\left| \begin{array}{cc}
a & b & c \\
d & e & f \\
g & h & i
\end{array} \right|
=
a(ei-hf)-b(di-gf)+c(dh-ge)\\
|A^1|
&=&
\left| \begin{array}{cc}
g & h & i \\
d & e & f \\
a & b & c
\end{array} \right|
=
a(hf-ie)-b(gf-id)+c(ge-hd)
=
-(a(-hf+ie)-b(-gf+id)+c(-ge+hd))
\\
|A^2|
&=&
\left| \begin{array}{cc}
g & h & i \\
a & b & c \\
d & e & f
\end{array} \right|
=
-a(hf-ie)+b(gf-id)-c(ge-hd)
=
a(ie-hf)-b(id-gf)+c(hd-ge)
\end{eqnarray}\]

\[\begin{eqnarray}
|A|
&=&
\left| \begin{array}{cc} a & b \\ c & d \end{array} \right|
= ad-bc \\
|A_{[1]*2}|
&=&
\left| \begin{array}{cc} 2a & 2b \\ c & d \end{array} \right|
= 2ad-2bc=2(ad-bc)
\end{eqnarray}\]

\[\begin{eqnarray}
|A_{[1]*2}|
&=&
\left| \begin{array}{cc}
2a & 2b & 2c \\
d & e & f \\
g & h & i
\end{array} \right|
=
2a(ei-hf)-2b(di-gf)+2c(dh-ge)
=
2(a(ei-hf)-b(di-gf)+c(dh-ge))\\
|A_{[2]*3}|
&=&
\left| \begin{array}{cc}
a & b & c \\
3d & 3e & 3f \\
g & h & i
\end{array} \right|
=
-3d(bi-ch)+3e(ai-cg)-3f(ah-bg)
=
3(-d(bi-ch)+e(ai-cg)-f(ah-bg))
\end{eqnarray}\]

\[\begin{eqnarray}
|A|
=
\left| \begin{array}{cc} a & b \\ c+2a & d+2b \end{array} \right|
&=&
a(d+2b)-b(c+2a)\\
&=&
ad+2ab-bc-2ab\\
&=&
ad-bc
\end{eqnarray}\]

\[\begin{eqnarray}
|A|
=
\left| \begin{array}{cc} a+3b & b \\ c+3d & d \end{array} \right|
&=&
(a+3b)d-b(c+3d)\\
&=&
ad+3bd-bc-3bd\\
&=&
ad-bc
\end{eqnarray}\]

\[\begin{eqnarray}
|A|
&=&
\left| \begin{array}{cc}
a & b & c \\
d+2a & e+2b & f+2c \\
g & h & i
\end{array} \right|\\
&=&
-(d+2a)(bi-ch)+(e+2b)(ai-cg)-(f+2c)(ah-bg)\\
&=&
aei-afh-bdi+bfg+cdh-ceg
\end{eqnarray}\]

\[\begin{eqnarray}
|A|
&=&
\left| \begin{array}{cc}
a+2g & b+2h & c+2i \\
d & e & f \\
g & h & i
\end{array} \right|\\
&=&
(a+2g)(ei-fh)-(b+2h)(di-fg)+(c+2i)(dh-eg)\\
&=&
aei-afh-bdi+bfg+cdh-ceg
\end{eqnarray}\]

\[\begin{eqnarray}
|A|
&=&
\left| \begin{array}{cc} a & b \\ a & b \end{array} \right|
= ab-ba
=0
\end{eqnarray}\]

\[\begin{eqnarray}
|A|
&=&
\left| \begin{array}{cc} a & a \\ b & b \end{array} \right|
= ab-ab
=0
\end{eqnarray}\]

\[\begin{eqnarray}
|A|
=
\left| \begin{array}{cc}
a & b & c \\
d & e & f \\
a & b & c
\end{array} \right|
&=&
a(ec-fb)-b(dc-fa)+c(db-ea)\\
&=&
aec-afb-bdc+afb+bdc-aec\\
&=&
0
\end{eqnarray}\]

\[\begin{eqnarray}
|A|
=
\left| \begin{array}{cc}
a & a & c \\
d & d & f \\
g & g & i
\end{array} \right|
&=&
a(di-fg)-d(ai-cg)+g(af-cd)\\
&=&
adi-afg-adi+cdg+afg-cdg\\
&=&
0
\end{eqnarray}\]