Для заданных матриц A и B решить матричное уравнение A X  B  E, где Е – единичная матрица.A    25 01    , B     31 4 2  

Для решения матричного уравнения A⋅X⋅B = E сначала найдем обратные матрицы A^(-1) и B^(-1).

A = [[25, -1], [0, 1]] det(A) = 251 - 0(-1) = 25 A^(-1) = (1/det(A)) * adj(A), где adj(A) - алгебраическое дополнение матрицы A, транспонированное:

adj(A) = [[1, 0], [0, 25]] A^(-1) = (1/25) * [[1, 0], [0, 25]] = [[1/25, 0], [0, 1/25]]

B = [[3, 1, 4], [2, 0, 1]] det(B) = 3*(01 - 12) - 1*(21 - 13) + 4*(21 - 03) = 3*(-2) - 1*(-1) + 4*2 = -6 + 1 + 8 = 3 B^(-1) = (1/det(B)) * adj(B), где adj(B) - алгебраическое дополнение матрицы B, транспонированное:

adj(B) = [[0, -3, 1], [-1, 3, -2], [2, 1, 0]] B^(-1) = (1/3) * [[0, -3, 1], [-1, 3, -2], [2, 1, 0]] = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]

Теперь подставим найденные обратные матрицы в уравнение A⋅X⋅B = E:

[[25, -1], [0, 1]]⋅X⋅[[3, 1, 4], [2, 0, 1]] = E [[25, -1], [0, 1]]⋅X⋅[[3, 1, 4], [2, 0, 1]] = [[1, 0, 0], [0, 1, 0]]

Умножим матрицы слева:

[[253 + 02, 251 + 00, 254 + (-1)1], [03 + 12, 01 + 10, 04 + 11]]⋅X = [[1, 0, 0], [0, 1, 0]] [[75, 25, 99], [2, 0, 1]]⋅X = [[1, 0, 0], [0, 1, 0]]

Теперь умножим обратную матрицу B^(-1) слева:

[[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]⋅[[75, 25, 99], [2, 0, 1]]⋅X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]⋅[[1, 0, 0], [0, 1, 0]]

[[(-1)25 + 02 + 1/399, (-1)0 + 00 + 1/31, (-1)4 + 11 + 1/30], [(-1/3)25 + 12 + (-2/3)99, (-1/3)0 + 10 + (-2/3)1, (-1/3)4 + 11 + (-2/3)0], [2/325 + 1/32 + 099, 2/30 + 1/30 + 01, 2/34 + 1/31 + 0*0]]⋅X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]

[[-25 + 0 + 33, 0 + 0 + 1/3, -4 + 1 + 0], [-25/3 + 2 - 198, 0 + 0 - 2/3, -4/3 + 1 - 0], [50/3 + 2/3 + 0, 0 + 1/3 + 0, 8/3 + 1/3 + 0]]⋅X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]

[[-25 + 33, 1/3, -3], [-25/3 - 196, 0 - 2/3, -4/3 + 1], [50/3 + 2/3, 1/3, 8/3 + 1/3]]⋅X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]

[[8, 1/3, -3], [-221/3, -2/3, -1/3], [52/3, 1/3, 9/3]]⋅X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]

Теперь умножим обратную матрицу A^(-1) справа:

[[8, 1/3, -3], [-221/3, -2/3, -1/3], [52/3, 1/3, 9/3]]⋅[[1/25, 0], [0, 1/25]] = [[8/25, 1/75], [-221/75, -2/75], [52/75, 1/75]]

[[8/25, 1/75], [-221/75, -2/75], [52/75, 1/75]]⋅X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]

Умножим матрицы слева:

[[8/250 + 1/75(-1/3) + (-3)2/75, 8/25(-1) + 1/751 + (-3)1/75, 8/251/3 + 1/750 + (-3)0]] + [[-221/750 + (-2/75)(-1/3) + (-1/3)2/75, -221/75(-1) + (-2/75)1 + (-1/3)1/75, -221/751/3 + (-2/75)0 + (-1/3)0]] + [[52/750 + 1/75(-1/3) + 9/752/3, 52/75(-1) + 1/751 + 9/751/3, 52/751/3 + 1/750 + 9/75*0]]⋅X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]

[[(-3)/75 - 2/75 - 6/75, (-8)/25 + 1/75 - 3/75, 8/75]] + [[2/75 + 2/75 - 2/75, 221/75 - 2/75 - 1/75, (-221)/75]] + [[0, 1/75 - 1/75 + 0, 0]]⋅X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]

[[-11/75, -8/25 - 2/75 - 3/75, 8/75] + [2/75, 218/75 - 3/75, -221/75] + [0, 0, 0]]⋅X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]

[[-11/75, -33/25, 8/75] + [2/75, 215/75, -221/75] + [0, 0, 0]]⋅X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]

Теперь умножим матрицы слева:

[[-11/750 + (-33/25)(-1) + 8/752/3, -11/75(-1) + (-33/25)1 + 8/751/3, -11/751/3 + (-33/25)0 + 8/750] + [2/750 + 215/75*(-1) + (-221/75)2/3, 2/75(-1) + 215/751 + (-221/75)1/3, 2/751/3 + 215/750 + (-221/75)*0]]⋅X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]

[[33/25 + 16/25 + 16/25, 11/75 - 33/25 + 8/75, 0] + [0, 215/75 - 442/75, 0]]⋅X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]

[[65/25, -22/75, 0] + [0, -227/75, 0]]⋅X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]

[[13, -22/75, 0] + [0, -227/75, 0]]⋅X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]

Теперь умножим обратную матрицу слева:

[[13, -22/75, 0] + [0, -227/75, 0]]⋅[[1/25, 0], [0, 1/25]] = [[13/25, -22/1875], [0, -227/75]]

[[13/25, -22/1875], [0, -227/75]]⋅X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]

Умножим матрицы слева:

[[13/250 + (-22/1875)(-1/3), 13/25*(-1) + (-22/1875)1, 13/251/3 + (-22/1875)0], [00 + (-227/75)(-1/3), 0(-1) + (-227/75)1, 01/3 + (-227/75)*0]]⋅X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]

[[22/5625, -13/25, 13/75], [227/225, -227/75, 0]]⋅X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]

Теперь умножим матрицы слева:

[[22/56250 + (-13/25)(-1) + 13/752/3, 22/5625(-1) + (-13/25)1 + 13/751/3, 22/56251/3 + (-13/25)0 + 13/750], [227/2250 + (-227/75)(-1) + 02/3, 227/225*(-1) + (-227/75)1 + 01/3, 227/225*1/3 + (-227/75)0 + 00]]⋅X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]

[[26/75, -13/25 - 13/25 + 26/75, 0], [227/75, 227/75 - 227/75, 0]]⋅X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]

[[26/75, 0, 0], [227/75, 0, 0]]⋅X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]

Теперь решим полученное матричное уравнение:

[[26/75, 0, 0], [227/75, 0, 0]]⋅X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]

X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]⋅[[75/26, 0], [0, 75/227]]

X = [[0, -1, 1/3], [-1/3, 1, -2/3], [2/3, 1/3, 0]]⋅[[75/26, 0], [0, 75/227]]

X = [[075/26 + (-1)0, 0(-1) + 10, 01/3 + (-1)0], [(-1/3)75/26 + 10, (-1/3)(-1) + 10, (-1/3)1/3 + 10], [2/375/26 + 1/30, 2/3*(-1) + 1/30, 2/31/3 + 1/3*0]]

X = [[0, 0, 0], [0, 0, 0], [0, 0, 0]]

Таким образом, решение матричного уравнения A⋅X⋅B = E равно X = [[0, 0, 0], [0, 0, 0], [0, 0, 0]].