Даны матрицы: $$A = \begin{pmatrix} 1 & -1 & 2 \\ N & 4 & 0 \\ -2 & -N & 2 \end{pmatrix}$$, $$B = \begin{pmatrix} -1 & 2 & N \\ 1 & 0 & 4 \\ 5 & 1 & -N \end{pmatrix}$$. Найти: 1) $$2A-5B$$; 2) $$AB$$; 3) $$BA$$; 4) $$AB+BA$$; 5) $$A^2$$

Решение:

Даны матрицы:

$$A = \begin{pmatrix} 1 & -1 & 2 \\ N & 4 & 0 \\ -2 & -N & 2 \end{pmatrix}$$

$$B = \begin{pmatrix} -1 & 2 & N \\ 1 & 0 & 4 \\ 5 & 1 & -N \end{pmatrix}$$

1) $$2A-5B$$

$$2A = 2 \begin{pmatrix} 1 & -1 & 2 \\ N & 4 & 0 \\ -2 & -N & 2 \end{pmatrix} = \begin{pmatrix} 2 & -2 & 4 \\ 2N & 8 & 0 \\ -4 & -2N & 4 \end{pmatrix}$$

$$5B = 5 \begin{pmatrix} -1 & 2 & N \\ 1 & 0 & 4 \\ 5 & 1 & -N \end{pmatrix} = \begin{pmatrix} -5 & 10 & 5N \\ 5 & 0 & 20 \\ 25 & 5 & -5N \end{pmatrix}$$

$$2A-5B = \begin{pmatrix} 2 & -2 & 4 \\ 2N & 8 & 0 \\ -4 & -2N & 4 \end{pmatrix} - \begin{pmatrix} -5 & 10 & 5N \\ 5 & 0 & 20 \\ 25 & 5 & -5N \end{pmatrix} = \begin{pmatrix} 2-(-5) & -2-10 & 4-5N \\ 2N-5 & 8-0 & 0-20 \\ -4-25 & -2N-5 & 4-(-5N) \end{pmatrix} = \begin{pmatrix} 7 & -12 & 4-5N \\ 2N-5 & 8 & -20 \\ -29 & -2N-5 & 4+5N \end{pmatrix}$$

2) $$AB$$

$$AB = \begin{pmatrix} 1 & -1 & 2 \\ N & 4 & 0 \\ -2 & -N & 2 \end{pmatrix} \begin{pmatrix} -1 & 2 & N \\ 1 & 0 & 4 \\ 5 & 1 & -N \end{pmatrix}$$

Элемент $$(1,1)$$: $$(1)(-1) + (-1)(1) + (2)(5) = -1 - 1 + 10 = 8$$

Элемент $$(1,2)$$: $$(1)(2) + (-1)(0) + (2)(1) = 2 + 0 + 2 = 4$$

Элемент $$(1,3)$$: $$(1)(N) + (-1)(4) + (2)(-N) = N - 4 - 2N = -N - 4$$

Элемент $$(2,1)$$: $$(N)(-1) + (4)(1) + (0)(5) = -N + 4 + 0 = 4 - N$$

Элемент $$(2,2)$$: $$(N)(2) + (4)(0) + (0)(1) = 2N + 0 + 0 = 2N$$

Элемент $$(2,3)$$: $$(N)(N) + (4)(4) + (0)(-N) = N^2 + 16 + 0 = N^2 + 16$$

Элемент $$(3,1)$$: $$(-2)(-1) + (-N)(1) + (2)(5) = 2 - N + 10 = 12 - N$$

Элемент $$(3,2)$$: $$(-2)(2) + (-N)(0) + (2)(1) = -4 + 0 + 2 = -2$$

Элемент $$(3,3)$$: $$(-2)(N) + (-N)(4) + (2)(-N) = -2N - 4N - 2N = -8N$$

$$AB = \begin{pmatrix} 8 & 4 & -N-4 \\ 4-N & 2N & N^2+16 \\ 12-N & -2 & -8N \end{pmatrix}$$

3) $$BA$$

$$BA = \begin{pmatrix} -1 & 2 & N \\ 1 & 0 & 4 \\ 5 & 1 & -N \end{pmatrix} \begin{pmatrix} 1 & -1 & 2 \\ N & 4 & 0 \\ -2 & -N & 2 \end{pmatrix}$$

Элемент $$(1,1)$$: $$(-1)(1) + (2)(N) + (N)(-2) = -1 + 2N - 2N = -1$$

Элемент $$(1,2)$$: $$(-1)(-1) + (2)(4) + (N)(-N) = 1 + 8 - N^2 = 9 - N^2$$

Элемент $$(1,3)$$: $$(-1)(2) + (2)(0) + (N)(2) = -2 + 0 + 2N = 2N - 2$$

Элемент $$(2,1)$$: $$(1)(1) + (0)(N) + (4)(-2) = 1 + 0 - 8 = -7$$

Элемент $$(2,2)$$: $$(1)(-1) + (0)(4) + (4)(-N) = -1 + 0 - 4N = -1 - 4N$$

Элемент $$(2,3)$$: $$(1)(2) + (0)(0) + (4)(2) = 2 + 0 + 8 = 10$$

Элемент $$(3,1)$$: $$(5)(1) + (1)(N) + (-N)(-2) = 5 + N + 2N = 5 + 3N$$

Элемент $$(3,2)$$: $$(5)(-1) + (1)(4) + (-N)(-N) = -5 + 4 + N^2 = N^2 - 1$$

Элемент $$(3,3)$$: $$(5)(2) + (1)(0) + (-N)(2) = 10 + 0 - 2N = 10 - 2N$$

$$BA = \begin{pmatrix} -1 & 9-N^2 & 2N-2 \\ -7 & -1-4N & 10 \\ 5+3N & N^2-1 & 10-2N \end{pmatrix}$$

4) $$AB+BA$$

$$AB+BA = \begin{pmatrix} 8 & 4 & -N-4 \\ 4-N & 2N & N^2+16 \\ 12-N & -2 & -8N \end{pmatrix} + \begin{pmatrix} -1 & 9-N^2 & 2N-2 \\ -7 & -1-4N & 10 \\ 5+3N & N^2-1 & 10-2N \end{pmatrix}$$

$$= \begin{pmatrix} 8+(-1) & 4+(9-N^2) & (-N-4)+(2N-2) \\ (4-N)+(-7) & 2N+(-1-4N) & (N^2+16)+10 \\ (12-N)+(5+3N) & -2+(N^2-1) & -8N+(10-2N) \end{pmatrix}$$

$$= \begin{pmatrix} 7 & 13-N^2 & N-6 \\ -3-N & -1-2N & N^2+26 \\ 17+2N & N^2-3 & 10-10N \end{pmatrix}$$

5) $$A^2$$

$$A^2 = A \cdot A = \begin{pmatrix} 1 & -1 & 2 \\ N & 4 & 0 \\ -2 & -N & 2 \end{pmatrix} \begin{pmatrix} 1 & -1 & 2 \\ N & 4 & 0 \\ -2 & -N & 2 \end{pmatrix}$$

Элемент $$(1,1)$$: $$(1)(1) + (-1)(N) + (2)(-2) = 1 - N - 4 = -N - 3$$

Элемент $$(1,2)$$: $$(1)(-1) + (-1)(4) + (2)(-N) = -1 - 4 - 2N = -5 - 2N$$

Элемент $$(1,3)$$: $$(1)(2) + (-1)(0) + (2)(2) = 2 + 0 + 4 = 6$$

Элемент $$(2,1)$$: $$(N)(1) + (4)(N) + (0)(-2) = N + 4N + 0 = 5N$$

Элемент $$(2,2)$$: $$(N)(-1) + (4)(4) + (0)(-N) = -N + 16 + 0 = 16 - N$$

Элемент $$(2,3)$$: $$(N)(2) + (4)(0) + (0)(2) = 2N + 0 + 0 = 2N$$

Элемент $$(3,1)$$: $$(-2)(1) + (-N)(N) + (2)(-2) = -2 - N^2 - 4 = -N^2 - 6$$

Элемент $$(3,2)$$: $$(-2)(-1) + (-N)(4) + (2)(-N) = 2 - 4N - 2N = 2 - 6N$$

Элемент $$(3,3)$$: $$(-2)(2) + (-N)(0) + (2)(2) = -4 + 0 + 4 = 0$$

$$A^2 = \begin{pmatrix} -N-3 & -5-2N & 6 \\ 5N & 16-N & 2N \\ -N^2-6 & 2-6N & 0 \end{pmatrix}$$

Ответ:

1) $$2A-5B = \begin{pmatrix} 7 & -12 & 4-5N \\ 2N-5 & 8 & -20 \\ -29 & -2N-5 & 4+5N \end{pmatrix}$$

2) $$AB = \begin{pmatrix} 8 & 4 & -N-4 \\ 4-N & 2N & N^2+16 \\ 12-N & -2 & -8N \end{pmatrix}$$

3) $$BA = \begin{pmatrix} -1 & 9-N^2 & 2N-2 \\ -7 & -1-4N & 10 \\ 5+3N & N^2-1 & 10-2N \end{pmatrix}$$

4) $$AB+BA = \begin{pmatrix} 7 & 13-N^2 & N-6 \\ -3-N & -1-2N & N^2+26 \\ 17+2N & N^2-3 & 10-10N \end{pmatrix}$$

5) $$A^2 = \begin{pmatrix} -N-3 & -5-2N & 6 \\ 5N & 16-N & 2N \\ -N^2-6 & 2-6N & 0 \end{pmatrix}$$