20. $$ A, B $$ va $$ C $$ matritsalar berilgan. $$ (A B) C=A(B C) $$ ekanini tekshiring $$ A=\left[\begin{array}{cc}5 & 3 \ 2 & -1 \ 4 & 7\end{array}\right], \quad B=\left[\begin{array}{cc}4 & -3 \ -5 & 2\end{array}\right], \quad C=\left[\begin{array}{ll}3 & 5 \ 1 & 2\end{array}\right] $$

Question

UpStudy AI · Accepted Answer

Keltirilgan matritsalar $$ A $$, $$ B $$ va $$ C $$ uchun $$(AB)C = A(BC)$$ ekvivalentligini tekshirish uchun quyidagi qadamlarni bajaramiz.

Berilgan matritsalar:
$$
A = \begin{bmatrix}
5 & 3 \
2 & -1 \
4 & 7
\end{bmatrix}, \quad
B = \begin{bmatrix}
4 & -3 \
-5 & 2
\end{bmatrix}, \quad
C = \begin{bmatrix}
3 & 5 \
1 & 2
\end{bmatrix}
$$

### 1. $$ AB $$ matritsasini hisoblash:
$$ A $$ (3x2) va $$ B $$ (2x2) matritsalarini ko‘paytiramiz.
$$
AB = \begin{bmatrix}
5 \times 4 + 3 \times (-5) & 5 \times (-3) + 3 \times 2 \
2 \times 4 + (-1) \times (-5) & 2 \times (-3) + (-1) \times 2 \
4 \times 4 + 7 \times (-5) & 4 \times (-3) + 7 \times 2
\end{bmatrix} = \begin{bmatrix}
5 & -9 \
13 & -8 \
-19 & 2
\end{bmatrix}
$$

### 2. $$ (AB)C $$ matritsasini hisoblash:
$$ AB $$ (3x2) va $$ C $$ (2x2) matritsalarini ko‘paytiramiz.
$$
(AB)C = \begin{bmatrix}
5 \times 3 + (-9) \times 1 & 5 \times 5 + (-9) \times 2 \
13 \times 3 + (-8) \times 1 & 13 \times 5 + (-8) \times 2 \
-19 \times 3 + 2 \times 1 & -19 \times 5 + 2 \times 2
\end{bmatrix} = \begin{bmatrix}
6 & 7 \
31 & 49 \
-55 & -91
\end{bmatrix}
$$

### 3. $$ BC $$ matritsasini hisoblash:
$$ B $$ (2x2) va $$ C $$ (2x2) matritsalarini ko‘paytiramiz.
$$
BC = \begin{bmatrix}
4 \times 3 + (-3) \times 1 & 4 \times 5 + (-3) \times 2 \
-5 \times 3 + 2 \times 1 & -5 \times 5 + 2 \times 2
\end{bmatrix} = \begin{bmatrix}
9 & 14 \
-13 & -21
\end{bmatrix}
$$

### 4. $$ A(BC) $$ matritsasini hisoblash:
$$ A $$ (3x2) va $$ BC $$ (2x2) matritsalarini ko‘paytiramiz.
$$
A(BC) = \begin{bmatrix}
5 \times 9 + 3 \times (-13) & 5 \times 14 + 3 \times (-21) \
2 \times 9 + (-1) \times (-13) & 2 \times 14 + (-1) \times (-21) \
4 \times 9 + 7 \times (-13) & 4 \times 14 + 7 \times (-21)
\end{bmatrix} = \begin{bmatrix}
6 & 7 \
31 & 49 \
-55 & -91
\end{bmatrix}
$$

### Natija:
$$
(AB)C = \begin{bmatrix}
6 & 7 \
31 & 49 \
-55 & -91
\end{bmatrix} = A(BC)
$$

Shunday qilib, $$(AB)C = A(BC)$$ ekanligi tasdiqlandi. Bu matritsa ko‘paytirishining assotsiativ xususiyati bilan bog‘liq.
$$(AB)C = A(BC)$$ ekanligi tasdiqlandi.

20. \( A, B \) va \( C \) matritsalar berilgan. \( (A B) C=A(B C) \) ekanini tekshiring \[ A=\left[\begin{array}{cc}5 & 3 \\ 2 & -1 \\ 4 & 7\end{array}\right], \quad B=\left[\begin{array}{cc}4 & -3 \\ -5 & 2\end{array}\right], \quad C=\left[\begin{array}{ll}3 & 5 \\ 1 & 2\end{array}\right] \]

Solución de tutoría real

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Other Questions