## 2018年1月29日月曜日

### 数学 - Python - 線型代数 - 行列式 - 積の行列式(n次正方行列、零行列、行列式の積)

1. $\begin{array}{}\mathrm{det}\left(\begin{array}{cc}B& C\\ O& D\end{array}\right)\end{array}=\mathrm{det}\left(\left(\begin{array}{cc}{I}_{m}& C\\ O& D\end{array}\right)\left(\begin{array}{cc}B& O\\ O& {I}_{n}\end{array}\right)\right)\\ =\mathrm{det}\left(\begin{array}{cc}{I}_{m}& C\\ O& D\end{array}\right)\mathrm{det}\left(\begin{array}{cc}B& O\\ O& {I}_{n}\end{array}\right)\\ =\mathrm{det}D·\mathrm{det}B\\ =\mathrm{det}B·\mathrm{det}D$

また、

$\begin{array}{}\mathrm{det}\left(\begin{array}{cc}B& O\\ C& D\end{array}\right)\end{array}=\mathrm{det}\left(\left(\begin{array}{cc}B& O\\ C& {I}_{n}\end{array}\right)\left(\begin{array}{cc}{I}_{m}& O\\ O& D\end{array}\right)\right)\\ =\mathrm{det}\left(\begin{array}{cc}B& O\\ C& {I}_{n}\end{array}\right)·\mathrm{det}\left(\begin{array}{cc}{I}_{m}& O\\ O& D\end{array}\right)\\ =\mathrm{det}B·\mathrm{det}D$

（証明終）

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Matrix, I

for m in range(1, 4):
B = Matrix([[symbols(f'b{i + 1}{j + 1}') for j in range(m)]
for i in range(m)])
DB = B.det()

def f(i, j):
if i < m and j < m:
return symbols(f'b{i + 1}{j + 1}')
if i < m:
return symbols(f'c{i + 1}{j + 1 - m}')
if j < m:
return 0
return symbols(f'd{i + 1 - m}{j + 1 - m}')

def g(i, j):
if i < m and j < m:
return symbols(f'b{i + 1}{j + 1}')
if i < m:
return 0
if j < m:
return symbols(f'c{i + 1 - m}{j + 1}')
return symbols(f'd{i + 1 - m}{j + 1 - m}')

for n in range(1, 4):
for h in [f, g]:
D = Matrix([[symbols(f'd{i + 1}{j + 1}') for j in range(n)]
for i in range(n)])
A = Matrix([[h(i, j) for j in range(m + n)]
for i in range(m + n)])
DA = A.det()
DD = D.det()
DBD = DB * DD
for t in [A, B, D, DA, DB, DD, DBD, DA == DBD.expand()]:
pprint(t)
print()
print()
print()
print()


$./sample2.py ⎡b₁₁ c₁₁⎤ ⎢ ⎥ ⎣ 0 d₁₁⎦ [b₁₁] [d₁₁] b₁₁⋅d₁₁ b₁₁ d₁₁ b₁₁⋅d₁₁ True ⎡b₁₁ 0 ⎤ ⎢ ⎥ ⎣c₁₁ d₁₁⎦ [b₁₁] [d₁₁] b₁₁⋅d₁₁ b₁₁ d₁₁ b₁₁⋅d₁₁ True ⎡b₁₁ c₁₁ c₁₂⎤ ⎢ ⎥ ⎢ 0 d₁₁ d₁₂⎥ ⎢ ⎥ ⎣ 0 d₂₁ d₂₂⎦ [b₁₁] ⎡d₁₁ d₁₂⎤ ⎢ ⎥ ⎣d₂₁ d₂₂⎦ b₁₁⋅d₁₁⋅d₂₂ - b₁₁⋅d₁₂⋅d₂₁ b₁₁ d₁₁⋅d₂₂ - d₁₂⋅d₂₁ b₁₁⋅(d₁₁⋅d₂₂ - d₁₂⋅d₂₁) True ⎡b₁₁ 0 0 ⎤ ⎢ ⎥ ⎢c₁₁ d₁₁ d₁₂⎥ ⎢ ⎥ ⎣c₂₁ d₂₁ d₂₂⎦ [b₁₁] ⎡d₁₁ d₁₂⎤ ⎢ ⎥ ⎣d₂₁ d₂₂⎦ b₁₁⋅d₁₁⋅d₂₂ - b₁₁⋅d₁₂⋅d₂₁ b₁₁ d₁₁⋅d₂₂ - d₁₂⋅d₂₁ b₁₁⋅(d₁₁⋅d₂₂ - d₁₂⋅d₂₁) True ⎡b₁₁ c₁₁ c₁₂ c₁₃⎤ ⎢ ⎥ ⎢ 0 d₁₁ d₁₂ d₁₃⎥ ⎢ ⎥ ⎢ 0 d₂₁ d₂₂ d₂₃⎥ ⎢ ⎥ ⎣ 0 d₃₁ d₃₂ d₃₃⎦ [b₁₁] ⎡d₁₁ d₁₂ d₁₃⎤ ⎢ ⎥ ⎢d₂₁ d₂₂ d₂₃⎥ ⎢ ⎥ ⎣d₃₁ d₃₂ d₃₃⎦ b₁₁⋅d₁₁⋅d₂₂⋅d₃₃ - b₁₁⋅d₁₁⋅d₂₃⋅d₃₂ - b₁₁⋅d₁₂⋅d₂₁⋅d₃₃ + b₁₁⋅d₁₂⋅d₂₃⋅d₃₁ + b₁₁⋅d₁ ₃⋅d₂₁⋅d₃₂ - b₁₁⋅d₁₃⋅d₂₂⋅d₃₁ b₁₁ d₁₁⋅d₂₂⋅d₃₃ - d₁₁⋅d₂₃⋅d₃₂ - d₁₂⋅d₂₁⋅d₃₃ + d₁₂⋅d₂₃⋅d₃₁ + d₁₃⋅d₂₁⋅d₃₂ - d₁₃⋅d₂₂⋅ d₃₁ b₁₁⋅(d₁₁⋅d₂₂⋅d₃₃ - d₁₁⋅d₂₃⋅d₃₂ - d₁₂⋅d₂₁⋅d₃₃ + d₁₂⋅d₂₃⋅d₃₁ + d₁₃⋅d₂₁⋅d₃₂ - d₁₃ ⋅d₂₂⋅d₃₁) True ⎡b₁₁ 0 0 0 ⎤ ⎢ ⎥ ⎢c₁₁ d₁₁ d₁₂ d₁₃⎥ ⎢ ⎥ ⎢c₂₁ d₂₁ d₂₂ d₂₃⎥ ⎢ ⎥ ⎣c₃₁ d₃₁ d₃₂ d₃₃⎦ [b₁₁] ⎡d₁₁ d₁₂ d₁₃⎤ ⎢ ⎥ ⎢d₂₁ d₂₂ d₂₃⎥ ⎢ ⎥ ⎣d₃₁ d₃₂ d₃₃⎦ b₁₁⋅d₁₁⋅d₂₂⋅d₃₃ - b₁₁⋅d₁₁⋅d₂₃⋅d₃₂ - b₁₁⋅d₁₂⋅d₂₁⋅d₃₃ + b₁₁⋅d₁₂⋅d₂₃⋅d₃₁ + b₁₁⋅d₁ ₃⋅d₂₁⋅d₃₂ - b₁₁⋅d₁₃⋅d₂₂⋅d₃₁ b₁₁ d₁₁⋅d₂₂⋅d₃₃ - d₁₁⋅d₂₃⋅d₃₂ - d₁₂⋅d₂₁⋅d₃₃ + d₁₂⋅d₂₃⋅d₃₁ + d₁₃⋅d₂₁⋅d₃₂ - d₁₃⋅d₂₂⋅ d₃₁ b₁₁⋅(d₁₁⋅d₂₂⋅d₃₃ - d₁₁⋅d₂₃⋅d₃₂ - d₁₂⋅d₂₁⋅d₃₃ + d₁₂⋅d₂₃⋅d₃₁ + d₁₃⋅d₂₁⋅d₃₂ - d₁₃ ⋅d₂₂⋅d₃₁) True ⎡b₁₁ b₁₂ c₁₁⎤ ⎢ ⎥ ⎢b₂₁ b₂₂ c₂₁⎥ ⎢ ⎥ ⎣ 0 0 d₁₁⎦ ⎡b₁₁ b₁₂⎤ ⎢ ⎥ ⎣b₂₁ b₂₂⎦ [d₁₁] b₁₁⋅b₂₂⋅d₁₁ - b₁₂⋅b₂₁⋅d₁₁ b₁₁⋅b₂₂ - b₁₂⋅b₂₁ d₁₁ d₁₁⋅(b₁₁⋅b₂₂ - b₁₂⋅b₂₁) True ⎡b₁₁ b₁₂ 0 ⎤ ⎢ ⎥ ⎢b₂₁ b₂₂ 0 ⎥ ⎢ ⎥ ⎣c₁₁ c₁₂ d₁₁⎦ ⎡b₁₁ b₁₂⎤ ⎢ ⎥ ⎣b₂₁ b₂₂⎦ [d₁₁] b₁₁⋅b₂₂⋅d₁₁ - b₁₂⋅b₂₁⋅d₁₁ b₁₁⋅b₂₂ - b₁₂⋅b₂₁ d₁₁ d₁₁⋅(b₁₁⋅b₂₂ - b₁₂⋅b₂₁) True ⎡b₁₁ b₁₂ c₁₁ c₁₂⎤ ⎢ ⎥ ⎢b₂₁ b₂₂ c₂₁ c₂₂⎥ ⎢ ⎥ ⎢ 0 0 d₁₁ d₁₂⎥ ⎢ ⎥ ⎣ 0 0 d₂₁ d₂₂⎦ ⎡b₁₁ b₁₂⎤ ⎢ ⎥ ⎣b₂₁ b₂₂⎦ ⎡d₁₁ d₁₂⎤ ⎢ ⎥ ⎣d₂₁ d₂₂⎦ b₁₁⋅b₂₂⋅d₁₁⋅d₂₂ - b₁₁⋅b₂₂⋅d₁₂⋅d₂₁ - b₁₂⋅b₂₁⋅d₁₁⋅d₂₂ + b₁₂⋅b₂₁⋅d₁₂⋅d₂₁ b₁₁⋅b₂₂ - b₁₂⋅b₂₁ d₁₁⋅d₂₂ - d₁₂⋅d₂₁ (b₁₁⋅b₂₂ - b₁₂⋅b₂₁)⋅(d₁₁⋅d₂₂ - d₁₂⋅d₂₁) True ⎡b₁₁ b₁₂ 0 0 ⎤ ⎢ ⎥ ⎢b₂₁ b₂₂ 0 0 ⎥ ⎢ ⎥ ⎢c₁₁ c₁₂ d₁₁ d₁₂⎥ ⎢ ⎥ ⎣c₂₁ c₂₂ d₂₁ d₂₂⎦ ⎡b₁₁ b₁₂⎤ ⎢ ⎥ ⎣b₂₁ b₂₂⎦ ⎡d₁₁ d₁₂⎤ ⎢ ⎥ ⎣d₂₁ d₂₂⎦ b₁₁⋅b₂₂⋅d₁₁⋅d₂₂ - b₁₁⋅b₂₂⋅d₁₂⋅d₂₁ - b₁₂⋅b₂₁⋅d₁₁⋅d₂₂ + b₁₂⋅b₂₁⋅d₁₂⋅d₂₁ b₁₁⋅b₂₂ - b₁₂⋅b₂₁ d₁₁⋅d₂₂ - d₁₂⋅d₂₁ (b₁₁⋅b₂₂ - b₁₂⋅b₂₁)⋅(d₁₁⋅d₂₂ - d₁₂⋅d₂₁) True ⎡b₁₁ b₁₂ c₁₁ c₁₂ c₁₃⎤ ⎢ ⎥ ⎢b₂₁ b₂₂ c₂₁ c₂₂ c₂₃⎥ ⎢ ⎥ ⎢ 0 0 d₁₁ d₁₂ d₁₃⎥ ⎢ ⎥ ⎢ 0 0 d₂₁ d₂₂ d₂₃⎥ ⎢ ⎥ ⎣ 0 0 d₃₁ d₃₂ d₃₃⎦ ⎡b₁₁ b₁₂⎤ ⎢ ⎥ ⎣b₂₁ b₂₂⎦ ⎡d₁₁ d₁₂ d₁₃⎤ ⎢ ⎥ ⎢d₂₁ d₂₂ d₂₃⎥ ⎢ ⎥ ⎣d₃₁ d₃₂ d₃₃⎦ b₁₁⋅b₂₂⋅d₁₁⋅d₂₂⋅d₃₃ - b₁₁⋅b₂₂⋅d₁₁⋅d₂₃⋅d₃₂ - b₁₁⋅b₂₂⋅d₁₂⋅d₂₁⋅d₃₃ + b₁₁⋅b₂₂⋅d₁₂⋅ d₂₃⋅d₃₁ + b₁₁⋅b₂₂⋅d₁₃⋅d₂₁⋅d₃₂ - b₁₁⋅b₂₂⋅d₁₃⋅d₂₂⋅d₃₁ - b₁₂⋅b₂₁⋅d₁₁⋅d₂₂⋅d₃₃ + b₁ ₂⋅b₂₁⋅d₁₁⋅d₂₃⋅d₃₂ + b₁₂⋅b₂₁⋅d₁₂⋅d₂₁⋅d₃₃ - b₁₂⋅b₂₁⋅d₁₂⋅d₂₃⋅d₃₁ - b₁₂⋅b₂₁⋅d₁₃⋅d₂ ₁⋅d₃₂ + b₁₂⋅b₂₁⋅d₁₃⋅d₂₂⋅d₃₁ b₁₁⋅b₂₂ - b₁₂⋅b₂₁ d₁₁⋅d₂₂⋅d₃₃ - d₁₁⋅d₂₃⋅d₃₂ - d₁₂⋅d₂₁⋅d₃₃ + d₁₂⋅d₂₃⋅d₃₁ + d₁₃⋅d₂₁⋅d₃₂ - d₁₃⋅d₂₂⋅ d₃₁ (b₁₁⋅b₂₂ - b₁₂⋅b₂₁)⋅(d₁₁⋅d₂₂⋅d₃₃ - d₁₁⋅d₂₃⋅d₃₂ - d₁₂⋅d₂₁⋅d₃₃ + d₁₂⋅d₂₃⋅d₃₁ + d ₁₃⋅d₂₁⋅d₃₂ - d₁₃⋅d₂₂⋅d₃₁) True ⎡b₁₁ b₁₂ 0 0 0 ⎤ ⎢ ⎥ ⎢b₂₁ b₂₂ 0 0 0 ⎥ ⎢ ⎥ ⎢c₁₁ c₁₂ d₁₁ d₁₂ d₁₃⎥ ⎢ ⎥ ⎢c₂₁ c₂₂ d₂₁ d₂₂ d₂₃⎥ ⎢ ⎥ ⎣c₃₁ c₃₂ d₃₁ d₃₂ d₃₃⎦ ⎡b₁₁ b₁₂⎤ ⎢ ⎥ ⎣b₂₁ b₂₂⎦ ⎡d₁₁ d₁₂ d₁₃⎤ ⎢ ⎥ ⎢d₂₁ d₂₂ d₂₃⎥ ⎢ ⎥ ⎣d₃₁ d₃₂ d₃₃⎦ b₁₁⋅b₂₂⋅d₁₁⋅d₂₂⋅d₃₃ - b₁₁⋅b₂₂⋅d₁₁⋅d₂₃⋅d₃₂ - b₁₁⋅b₂₂⋅d₁₂⋅d₂₁⋅d₃₃ + b₁₁⋅b₂₂⋅d₁₂⋅ d₂₃⋅d₃₁ + b₁₁⋅b₂₂⋅d₁₃⋅d₂₁⋅d₃₂ - b₁₁⋅b₂₂⋅d₁₃⋅d₂₂⋅d₃₁ - b₁₂⋅b₂₁⋅d₁₁⋅d₂₂⋅d₃₃ + b₁ ₂⋅b₂₁⋅d₁₁⋅d₂₃⋅d₃₂ + b₁₂⋅b₂₁⋅d₁₂⋅d₂₁⋅d₃₃ - b₁₂⋅b₂₁⋅d₁₂⋅d₂₃⋅d₃₁ - b₁₂⋅b₂₁⋅d₁₃⋅d₂ ₁⋅d₃₂ + b₁₂⋅b₂₁⋅d₁₃⋅d₂₂⋅d₃₁ b₁₁⋅b₂₂ - b₁₂⋅b₂₁ d₁₁⋅d₂₂⋅d₃₃ - d₁₁⋅d₂₃⋅d₃₂ - d₁₂⋅d₂₁⋅d₃₃ + d₁₂⋅d₂₃⋅d₃₁ + d₁₃⋅d₂₁⋅d₃₂ - d₁₃⋅d₂₂⋅ d₃₁ (b₁₁⋅b₂₂ - b₁₂⋅b₂₁)⋅(d₁₁⋅d₂₂⋅d₃₃ - d₁₁⋅d₂₃⋅d₃₂ - d₁₂⋅d₂₁⋅d₃₃ + d₁₂⋅d₂₃⋅d₃₁ + d ₁₃⋅d₂₁⋅d₃₂ - d₁₃⋅d₂₂⋅d₃₁) True ⎡b₁₁ b₁₂ b₁₃ c₁₁⎤ ⎢ ⎥ ⎢b₂₁ b₂₂ b₂₃ c₂₁⎥ ⎢ ⎥ ⎢b₃₁ b₃₂ b₃₃ c₃₁⎥ ⎢ ⎥ ⎣ 0 0 0 d₁₁⎦ ⎡b₁₁ b₁₂ b₁₃⎤ ⎢ ⎥ ⎢b₂₁ b₂₂ b₂₃⎥ ⎢ ⎥ ⎣b₃₁ b₃₂ b₃₃⎦ [d₁₁] b₁₁⋅b₂₂⋅b₃₃⋅d₁₁ - b₁₁⋅b₂₃⋅b₃₂⋅d₁₁ - b₁₂⋅b₂₁⋅b₃₃⋅d₁₁ + b₁₂⋅b₂₃⋅b₃₁⋅d₁₁ + b₁₃⋅b₂ ₁⋅b₃₂⋅d₁₁ - b₁₃⋅b₂₂⋅b₃₁⋅d₁₁ b₁₁⋅b₂₂⋅b₃₃ - b₁₁⋅b₂₃⋅b₃₂ - b₁₂⋅b₂₁⋅b₃₃ + b₁₂⋅b₂₃⋅b₃₁ + b₁₃⋅b₂₁⋅b₃₂ - b₁₃⋅b₂₂⋅ b₃₁ d₁₁ d₁₁⋅(b₁₁⋅b₂₂⋅b₃₃ - b₁₁⋅b₂₃⋅b₃₂ - b₁₂⋅b₂₁⋅b₃₃ + b₁₂⋅b₂₃⋅b₃₁ + b₁₃⋅b₂₁⋅b₃₂ - b₁₃ ⋅b₂₂⋅b₃₁) True ⎡b₁₁ b₁₂ b₁₃ 0 ⎤ ⎢ ⎥ ⎢b₂₁ b₂₂ b₂₃ 0 ⎥ ⎢ ⎥ ⎢b₃₁ b₃₂ b₃₃ 0 ⎥ ⎢ ⎥ ⎣c₁₁ c₁₂ c₁₃ d₁₁⎦ ⎡b₁₁ b₁₂ b₁₃⎤ ⎢ ⎥ ⎢b₂₁ b₂₂ b₂₃⎥ ⎢ ⎥ ⎣b₃₁ b₃₂ b₃₃⎦ [d₁₁] b₁₁⋅b₂₂⋅b₃₃⋅d₁₁ - b₁₁⋅b₂₃⋅b₃₂⋅d₁₁ - b₁₂⋅b₂₁⋅b₃₃⋅d₁₁ + b₁₂⋅b₂₃⋅b₃₁⋅d₁₁ + b₁₃⋅b₂ ₁⋅b₃₂⋅d₁₁ - b₁₃⋅b₂₂⋅b₃₁⋅d₁₁ b₁₁⋅b₂₂⋅b₃₃ - b₁₁⋅b₂₃⋅b₃₂ - b₁₂⋅b₂₁⋅b₃₃ + b₁₂⋅b₂₃⋅b₃₁ + b₁₃⋅b₂₁⋅b₃₂ - b₁₃⋅b₂₂⋅ b₃₁ d₁₁ d₁₁⋅(b₁₁⋅b₂₂⋅b₃₃ - b₁₁⋅b₂₃⋅b₃₂ - b₁₂⋅b₂₁⋅b₃₃ + b₁₂⋅b₂₃⋅b₃₁ + b₁₃⋅b₂₁⋅b₃₂ - b₁₃ ⋅b₂₂⋅b₃₁) True ⎡b₁₁ b₁₂ b₁₃ c₁₁ c₁₂⎤ ⎢ ⎥ ⎢b₂₁ b₂₂ b₂₃ c₂₁ c₂₂⎥ ⎢ ⎥ ⎢b₃₁ b₃₂ b₃₃ c₃₁ c₃₂⎥ ⎢ ⎥ ⎢ 0 0 0 d₁₁ d₁₂⎥ ⎢ ⎥ ⎣ 0 0 0 d₂₁ d₂₂⎦ ⎡b₁₁ b₁₂ b₁₃⎤ ⎢ ⎥ ⎢b₂₁ b₂₂ b₂₃⎥ ⎢ ⎥ ⎣b₃₁ b₃₂ b₃₃⎦ ⎡d₁₁ d₁₂⎤ ⎢ ⎥ ⎣d₂₁ d₂₂⎦ b₁₁⋅b₂₂⋅b₃₃⋅d₁₁⋅d₂₂ - b₁₁⋅b₂₂⋅b₃₃⋅d₁₂⋅d₂₁ - b₁₁⋅b₂₃⋅b₃₂⋅d₁₁⋅d₂₂ + b₁₁⋅b₂₃⋅b₃₂⋅ d₁₂⋅d₂₁ - b₁₂⋅b₂₁⋅b₃₃⋅d₁₁⋅d₂₂ + b₁₂⋅b₂₁⋅b₃₃⋅d₁₂⋅d₂₁ + b₁₂⋅b₂₃⋅b₃₁⋅d₁₁⋅d₂₂ - b₁ ₂⋅b₂₃⋅b₃₁⋅d₁₂⋅d₂₁ + b₁₃⋅b₂₁⋅b₃₂⋅d₁₁⋅d₂₂ - b₁₃⋅b₂₁⋅b₃₂⋅d₁₂⋅d₂₁ - b₁₃⋅b₂₂⋅b₃₁⋅d₁ ₁⋅d₂₂ + b₁₃⋅b₂₂⋅b₃₁⋅d₁₂⋅d₂₁ b₁₁⋅b₂₂⋅b₃₃ - b₁₁⋅b₂₃⋅b₃₂ - b₁₂⋅b₂₁⋅b₃₃ + b₁₂⋅b₂₃⋅b₃₁ + b₁₃⋅b₂₁⋅b₃₂ - b₁₃⋅b₂₂⋅ b₃₁ d₁₁⋅d₂₂ - d₁₂⋅d₂₁ (d₁₁⋅d₂₂ - d₁₂⋅d₂₁)⋅(b₁₁⋅b₂₂⋅b₃₃ - b₁₁⋅b₂₃⋅b₃₂ - b₁₂⋅b₂₁⋅b₃₃ + b₁₂⋅b₂₃⋅b₃₁ + b ₁₃⋅b₂₁⋅b₃₂ - b₁₃⋅b₂₂⋅b₃₁) True ⎡b₁₁ b₁₂ b₁₃ 0 0 ⎤ ⎢ ⎥ ⎢b₂₁ b₂₂ b₂₃ 0 0 ⎥ ⎢ ⎥ ⎢b₃₁ b₃₂ b₃₃ 0 0 ⎥ ⎢ ⎥ ⎢c₁₁ c₁₂ c₁₃ d₁₁ d₁₂⎥ ⎢ ⎥ ⎣c₂₁ c₂₂ c₂₃ d₂₁ d₂₂⎦ ⎡b₁₁ b₁₂ b₁₃⎤ ⎢ ⎥ ⎢b₂₁ b₂₂ b₂₃⎥ ⎢ ⎥ ⎣b₃₁ b₃₂ b₃₃⎦ ⎡d₁₁ d₁₂⎤ ⎢ ⎥ ⎣d₂₁ d₂₂⎦ b₁₁⋅b₂₂⋅b₃₃⋅d₁₁⋅d₂₂ - b₁₁⋅b₂₂⋅b₃₃⋅d₁₂⋅d₂₁ - b₁₁⋅b₂₃⋅b₃₂⋅d₁₁⋅d₂₂ + b₁₁⋅b₂₃⋅b₃₂⋅ d₁₂⋅d₂₁ - b₁₂⋅b₂₁⋅b₃₃⋅d₁₁⋅d₂₂ + b₁₂⋅b₂₁⋅b₃₃⋅d₁₂⋅d₂₁ + b₁₂⋅b₂₃⋅b₃₁⋅d₁₁⋅d₂₂ - b₁ ₂⋅b₂₃⋅b₃₁⋅d₁₂⋅d₂₁ + b₁₃⋅b₂₁⋅b₃₂⋅d₁₁⋅d₂₂ - b₁₃⋅b₂₁⋅b₃₂⋅d₁₂⋅d₂₁ - b₁₃⋅b₂₂⋅b₃₁⋅d₁ ₁⋅d₂₂ + b₁₃⋅b₂₂⋅b₃₁⋅d₁₂⋅d₂₁ b₁₁⋅b₂₂⋅b₃₃ - b₁₁⋅b₂₃⋅b₃₂ - b₁₂⋅b₂₁⋅b₃₃ + b₁₂⋅b₂₃⋅b₃₁ + b₁₃⋅b₂₁⋅b₃₂ - b₁₃⋅b₂₂⋅ b₃₁ d₁₁⋅d₂₂ - d₁₂⋅d₂₁ (d₁₁⋅d₂₂ - d₁₂⋅d₂₁)⋅(b₁₁⋅b₂₂⋅b₃₃ - b₁₁⋅b₂₃⋅b₃₂ - b₁₂⋅b₂₁⋅b₃₃ + b₁₂⋅b₂₃⋅b₃₁ + b ₁₃⋅b₂₁⋅b₃₂ - b₁₃⋅b₂₂⋅b₃₁) True ⎡b₁₁ b₁₂ b₁₃ c₁₁ c₁₂ c₁₃⎤ ⎢ ⎥ ⎢b₂₁ b₂₂ b₂₃ c₂₁ c₂₂ c₂₃⎥ ⎢ ⎥ ⎢b₃₁ b₃₂ b₃₃ c₃₁ c₃₂ c₃₃⎥ ⎢ ⎥ ⎢ 0 0 0 d₁₁ d₁₂ d₁₃⎥ ⎢ ⎥ ⎢ 0 0 0 d₂₁ d₂₂ d₂₃⎥ ⎢ ⎥ ⎣ 0 0 0 d₃₁ d₃₂ d₃₃⎦ ⎡b₁₁ b₁₂ b₁₃⎤ ⎢ ⎥ ⎢b₂₁ b₂₂ b₂₃⎥ ⎢ ⎥ ⎣b₃₁ b₃₂ b₃₃⎦ ⎡d₁₁ d₁₂ d₁₃⎤ ⎢ ⎥ ⎢d₂₁ d₂₂ d₂₃⎥ ⎢ ⎥ ⎣d₃₁ d₃₂ d₃₃⎦ b₁₁⋅b₂₂⋅b₃₃⋅d₁₁⋅d₂₂⋅d₃₃ - b₁₁⋅b₂₂⋅b₃₃⋅d₁₁⋅d₂₃⋅d₃₂ - b₁₁⋅b₂₂⋅b₃₃⋅d₁₂⋅d₂₁⋅d₃₃ + b₁₁⋅b₂₂⋅b₃₃⋅d₁₂⋅d₂₃⋅d₃₁ + b₁₁⋅b₂₂⋅b₃₃⋅d₁₃⋅d₂₁⋅d₃₂ - b₁₁⋅b₂₂⋅b₃₃⋅d₁₃⋅d₂₂⋅d₃₁ - b₁₁⋅b₂₃⋅b₃₂⋅d₁₁⋅d₂₂⋅d₃₃ + b₁₁⋅b₂₃⋅b₃₂⋅d₁₁⋅d₂₃⋅d₃₂ + b₁₁⋅b₂₃⋅b₃₂⋅d₁₂⋅d₂₁⋅d₃₃ - b₁₁⋅b₂₃⋅b₃₂⋅d₁₂⋅d₂₃⋅d₃₁ - b₁₁⋅b₂₃⋅b₃₂⋅d₁₃⋅d₂₁⋅d₃₂ + b₁₁⋅b₂₃⋅b₃₂⋅d₁₃⋅d₂₂⋅d₃₁ - b₁₂⋅b₂₁⋅b₃₃⋅d₁₁⋅d₂₂⋅d₃₃ + b₁₂⋅b₂₁⋅b₃₃⋅d₁₁⋅d₂₃⋅d₃₂ + b₁₂⋅b₂₁⋅b₃₃⋅d₁₂⋅d₂₁⋅d₃₃ - b₁₂⋅b₂₁⋅b₃₃⋅d₁₂⋅d₂₃⋅d₃₁ - b₁₂⋅b₂₁⋅b₃₃⋅d₁₃⋅d₂₁⋅d₃₂ + b₁₂⋅b₂₁⋅b₃₃⋅d₁₃⋅d₂₂⋅d₃₁ + b₁₂⋅b₂₃⋅b₃₁⋅d₁₁⋅d₂₂⋅d₃₃ - b₁₂⋅b₂₃⋅b₃₁⋅d₁₁⋅d₂₃⋅d₃₂ - b₁₂⋅b₂₃⋅b₃₁⋅d₁₂⋅d₂₁⋅d₃₃ + b₁₂⋅b₂₃⋅b₃₁⋅d₁₂⋅d₂₃⋅d₃₁ + b₁₂⋅b₂₃⋅b₃₁⋅d₁₃⋅d₂₁⋅d₃₂ - b₁₂⋅b₂₃⋅b₃₁⋅d₁₃⋅d₂₂⋅d₃₁ + b₁₃⋅b₂₁⋅b₃₂⋅d₁₁⋅d₂₂⋅d₃₃ - b₁₃⋅b₂₁⋅b₃₂⋅d₁₁⋅d₂₃⋅d₃₂ - b₁₃⋅b₂₁⋅b₃₂⋅d₁₂⋅d₂₁⋅d₃₃ + b₁₃⋅b₂₁⋅b₃₂⋅d₁₂⋅d₂₃⋅d₃₁ + b₁₃⋅b₂₁⋅b₃₂⋅d₁₃⋅d₂₁⋅d₃₂ - b₁₃⋅b₂₁⋅b₃₂⋅d₁₃⋅d₂₂⋅d₃₁ - b₁₃⋅b₂₂⋅b₃₁⋅d₁₁⋅d₂₂⋅d₃₃ + b₁₃⋅b₂₂⋅b₃₁⋅d₁₁⋅d₂₃⋅d₃₂ + b₁₃⋅b₂₂⋅b₃₁⋅d₁₂⋅d₂₁⋅d₃₃ - b₁₃⋅b₂₂⋅b₃₁⋅d₁₂⋅d₂₃⋅d₃₁ - b₁₃⋅b₂₂⋅b₃₁⋅d₁₃⋅d₂₁⋅d₃₂ + b₁₃⋅b₂₂⋅b₃₁⋅d₁₃⋅d₂₂⋅d₃₁ b₁₁⋅b₂₂⋅b₃₃ - b₁₁⋅b₂₃⋅b₃₂ - b₁₂⋅b₂₁⋅b₃₃ + b₁₂⋅b₂₃⋅b₃₁ + b₁₃⋅b₂₁⋅b₃₂ - b₁₃⋅b₂₂⋅ b₃₁ d₁₁⋅d₂₂⋅d₃₃ - d₁₁⋅d₂₃⋅d₃₂ - d₁₂⋅d₂₁⋅d₃₃ + d₁₂⋅d₂₃⋅d₃₁ + d₁₃⋅d₂₁⋅d₃₂ - d₁₃⋅d₂₂⋅ d₃₁ (b₁₁⋅b₂₂⋅b₃₃ - b₁₁⋅b₂₃⋅b₃₂ - b₁₂⋅b₂₁⋅b₃₃ + b₁₂⋅b₂₃⋅b₃₁ + b₁₃⋅b₂₁⋅b₃₂ - b₁₃⋅b₂₂ ⋅b₃₁)⋅(d₁₁⋅d₂₂⋅d₃₃ - d₁₁⋅d₂₃⋅d₃₂ - d₁₂⋅d₂₁⋅d₃₃ + d₁₂⋅d₂₃⋅d₃₁ + d₁₃⋅d₂₁⋅d₃₂ - d ₁₃⋅d₂₂⋅d₃₁) True ⎡b₁₁ b₁₂ b₁₃ 0 0 0 ⎤ ⎢ ⎥ ⎢b₂₁ b₂₂ b₂₃ 0 0 0 ⎥ ⎢ ⎥ ⎢b₃₁ b₃₂ b₃₃ 0 0 0 ⎥ ⎢ ⎥ ⎢c₁₁ c₁₂ c₁₃ d₁₁ d₁₂ d₁₃⎥ ⎢ ⎥ ⎢c₂₁ c₂₂ c₂₃ d₂₁ d₂₂ d₂₃⎥ ⎢ ⎥ ⎣c₃₁ c₃₂ c₃₃ d₃₁ d₃₂ d₃₃⎦ ⎡b₁₁ b₁₂ b₁₃⎤ ⎢ ⎥ ⎢b₂₁ b₂₂ b₂₃⎥ ⎢ ⎥ ⎣b₃₁ b₃₂ b₃₃⎦ ⎡d₁₁ d₁₂ d₁₃⎤ ⎢ ⎥ ⎢d₂₁ d₂₂ d₂₃⎥ ⎢ ⎥ ⎣d₃₁ d₃₂ d₃₃⎦ b₁₁⋅b₂₂⋅b₃₃⋅d₁₁⋅d₂₂⋅d₃₃ - b₁₁⋅b₂₂⋅b₃₃⋅d₁₁⋅d₂₃⋅d₃₂ - b₁₁⋅b₂₂⋅b₃₃⋅d₁₂⋅d₂₁⋅d₃₃ + b₁₁⋅b₂₂⋅b₃₃⋅d₁₂⋅d₂₃⋅d₃₁ + b₁₁⋅b₂₂⋅b₃₃⋅d₁₃⋅d₂₁⋅d₃₂ - b₁₁⋅b₂₂⋅b₃₃⋅d₁₃⋅d₂₂⋅d₃₁ - b₁₁⋅b₂₃⋅b₃₂⋅d₁₁⋅d₂₂⋅d₃₃ + b₁₁⋅b₂₃⋅b₃₂⋅d₁₁⋅d₂₃⋅d₃₂ + b₁₁⋅b₂₃⋅b₃₂⋅d₁₂⋅d₂₁⋅d₃₃ - b₁₁⋅b₂₃⋅b₃₂⋅d₁₂⋅d₂₃⋅d₃₁ - b₁₁⋅b₂₃⋅b₃₂⋅d₁₃⋅d₂₁⋅d₃₂ + b₁₁⋅b₂₃⋅b₃₂⋅d₁₃⋅d₂₂⋅d₃₁ - b₁₂⋅b₂₁⋅b₃₃⋅d₁₁⋅d₂₂⋅d₃₃ + b₁₂⋅b₂₁⋅b₃₃⋅d₁₁⋅d₂₃⋅d₃₂ + b₁₂⋅b₂₁⋅b₃₃⋅d₁₂⋅d₂₁⋅d₃₃ - b₁₂⋅b₂₁⋅b₃₃⋅d₁₂⋅d₂₃⋅d₃₁ - b₁₂⋅b₂₁⋅b₃₃⋅d₁₃⋅d₂₁⋅d₃₂ + b₁₂⋅b₂₁⋅b₃₃⋅d₁₃⋅d₂₂⋅d₃₁ + b₁₂⋅b₂₃⋅b₃₁⋅d₁₁⋅d₂₂⋅d₃₃ - b₁₂⋅b₂₃⋅b₃₁⋅d₁₁⋅d₂₃⋅d₃₂ - b₁₂⋅b₂₃⋅b₃₁⋅d₁₂⋅d₂₁⋅d₃₃ + b₁₂⋅b₂₃⋅b₃₁⋅d₁₂⋅d₂₃⋅d₃₁ + b₁₂⋅b₂₃⋅b₃₁⋅d₁₃⋅d₂₁⋅d₃₂ - b₁₂⋅b₂₃⋅b₃₁⋅d₁₃⋅d₂₂⋅d₃₁ + b₁₃⋅b₂₁⋅b₃₂⋅d₁₁⋅d₂₂⋅d₃₃ - b₁₃⋅b₂₁⋅b₃₂⋅d₁₁⋅d₂₃⋅d₃₂ - b₁₃⋅b₂₁⋅b₃₂⋅d₁₂⋅d₂₁⋅d₃₃ + b₁₃⋅b₂₁⋅b₃₂⋅d₁₂⋅d₂₃⋅d₃₁ + b₁₃⋅b₂₁⋅b₃₂⋅d₁₃⋅d₂₁⋅d₃₂ - b₁₃⋅b₂₁⋅b₃₂⋅d₁₃⋅d₂₂⋅d₃₁ - b₁₃⋅b₂₂⋅b₃₁⋅d₁₁⋅d₂₂⋅d₃₃ + b₁₃⋅b₂₂⋅b₃₁⋅d₁₁⋅d₂₃⋅d₃₂ + b₁₃⋅b₂₂⋅b₃₁⋅d₁₂⋅d₂₁⋅d₃₃ - b₁₃⋅b₂₂⋅b₃₁⋅d₁₂⋅d₂₃⋅d₃₁ - b₁₃⋅b₂₂⋅b₃₁⋅d₁₃⋅d₂₁⋅d₃₂ + b₁₃⋅b₂₂⋅b₃₁⋅d₁₃⋅d₂₂⋅d₃₁ b₁₁⋅b₂₂⋅b₃₃ - b₁₁⋅b₂₃⋅b₃₂ - b₁₂⋅b₂₁⋅b₃₃ + b₁₂⋅b₂₃⋅b₃₁ + b₁₃⋅b₂₁⋅b₃₂ - b₁₃⋅b₂₂⋅ b₃₁ d₁₁⋅d₂₂⋅d₃₃ - d₁₁⋅d₂₃⋅d₃₂ - d₁₂⋅d₂₁⋅d₃₃ + d₁₂⋅d₂₃⋅d₃₁ + d₁₃⋅d₂₁⋅d₃₂ - d₁₃⋅d₂₂⋅ d₃₁ (b₁₁⋅b₂₂⋅b₃₃ - b₁₁⋅b₂₃⋅b₃₂ - b₁₂⋅b₂₁⋅b₃₃ + b₁₂⋅b₂₃⋅b₃₁ + b₁₃⋅b₂₁⋅b₃₂ - b₁₃⋅b₂₂ ⋅b₃₁)⋅(d₁₁⋅d₂₂⋅d₃₃ - d₁₁⋅d₂₃⋅d₃₂ - d₁₂⋅d₂₁⋅d₃₃ + d₁₂⋅d₂₃⋅d₃₁ + d₁₃⋅d₂₁⋅d₃₂ - d ₁₃⋅d₂₂⋅d₃₁) True$