2018年5月28日月曜日

学習環境

ラング線形代数学(上)(S.ラング (著)、芹沢 正三 (翻訳)、ちくま学芸文庫)の6章(線形写像)、9(行列の逆転)、練習問題1-(a)、(b)、(c)、(d)、(e).を取り組んでみる。



    1. det A = 6 - 4 - - 2 + 24 = 2 - 22 = - 20 A 11 = 3 + 1 = 4 A 12 = 4 A 13 = - 12 A 21 = 1 - 2 = - 1 A 22 = 2 - 8 = - 6 A 23 = 2 - 4 = - 2 A 31 = - 1 - 6 = - 7 A 32 = - 2 A 33 = 6

      よって求める逆行列は、

      A - 1 = - 1 20 4 1 - 7 - 4 - 6 2 - 12 2 6

    2. det A = 18 + 2 - 20 - 12 + 3 - 20 = 5 A 11 = 6 - 4 = 2 A 12 = - 3 + 2 = - 1 A 13 = - 4 + 4 = 0 A 21 = - 3 - 20 = - 23 A 22 = 9 + 10 = 19 A 23 = 12 - 2 = 10 A 31 = - 1 - 10 = - 11 A 32 = 3 + 5 = 8 A 33 = 6 - 1 = 5

      よって 求める逆行列は、

      A - 1 = 1 5 2 23 - 11 1 19 - 8 0 - 10 5

    3. det A = 6 - 6 - - 4 = 4 A 11 = 3 A 12 = - 1 A 13 = - 2 A 21 = 4 - 6 = - 2 A 22 = 2 A 23 = 4 A 31 = - 9 A 32 = 3 A 33 = 6 + 4 = 10

      よって、求める逆行列は、

      A - 1 = 1 4 3 2 - 9 1 2 - 3 - 2 - 4 10

    4. det A = 7 - 2 = 5 A 11 = 7 - 2 = 5 A 12 = 0 A 13 = 0 A 21 = 14 + 2 = 16 A 22 = 7 A 23 = 2 A 31 = 2 + 1 = 3 A 32 = 1 A 33 = 1

      よって、

      A - 1 = 1 5 5 - 16 3 0 7 - 1 0 - 2 1

    5. det A = 84 - 160 = - 76 A 11 = 0 A 12 = 32 A 13 = 28 A 21 = 40 - 21 = 19 A 22 = - 8 - 6 = - 14 A 23 = - 7 - 10 = - 17 A 31 = 0 A 32 = - 12 A 33 = - 20

      よって、

      A - 1 = 1 - 76 0 - 19 0 - 32 - 14 12 28 17 - 20

コード(Emacs)

Python 3

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

print('1.')
ts = [[[2, 1, 2],
       [0, 3, -1],
       [4, 1, 1]],
      [[3, -1, 5],
       [-1, 2, 1],
       [-2, 4, 3]],
      [[2, 4, 3],
       [-1, 3, 0],
       [0, 2, 1]],
      [[1, 2, -1],
       [0, 1, 1],
       [0, 2, 7]],
      [[-1, 5, 3],
       [4, 0, 0],
       [2, 7, 8]]]

for i, t in enumerate(ts):
    print(f'({chr(ord("a") + i)})')
    A = Matrix(t)
    for M in [A, A**-1]:
        pprint(M)
        print()
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample1.py
1.
(a)
⎡2  1  2 ⎤
⎢        ⎥
⎢0  3  -1⎥
⎢        ⎥
⎣4  1  1 ⎦

⎡-1/5  -1/20  7/20 ⎤
⎢                  ⎥
⎢1/5   3/10   -1/10⎥
⎢                  ⎥
⎣3/5   -1/10  -3/10⎦


(b)
⎡3   -1  5⎤
⎢         ⎥
⎢-1  2   1⎥
⎢         ⎥
⎣-2  4   3⎦

⎡2/5  23/5  -11/5⎤
⎢                ⎥
⎢1/5  19/5  -8/5 ⎥
⎢                ⎥
⎣ 0    -2     1  ⎦


(c)
⎡2   4  3⎤
⎢        ⎥
⎢-1  3  0⎥
⎢        ⎥
⎣0   2  1⎦

⎡3/4   1/2  -9/4⎤
⎢               ⎥
⎢1/4   1/2  -3/4⎥
⎢               ⎥
⎣-1/2  -1   5/2 ⎦


(d)
⎡1  2  -1⎤
⎢        ⎥
⎢0  1  1 ⎥
⎢        ⎥
⎣0  2  7 ⎦

⎡1  -16/5  3/5 ⎤
⎢              ⎥
⎢0   7/5   -1/5⎥
⎢              ⎥
⎣0  -2/5   1/5 ⎦


(e)
⎡-1  5  3⎤
⎢        ⎥
⎢4   0  0⎥
⎢        ⎥
⎣2   7  8⎦

⎡  0    1/4     0  ⎤
⎢                  ⎥
⎢8/19   7/38  -3/19⎥
⎢                  ⎥
⎢       -17        ⎥
⎢-7/19  ────  5/19 ⎥
⎣        76        ⎦


$

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