Lang Ball
02/16/2023 · Senior High School
1. Diketahui \( A=\left[\begin{array}{lll}2 & 1 & 3 \\ -4 & 5 & 2 \\ 3 & 4 & 1\end{array}\right] \quad \) dan \( B=\left[\begin{array}{lll}-5 & 2 & 3 \\ 2 & 4 & -5 \\ 1 & 3 & 4\end{array}\right] \) Tentukan hasil dari a. \( 3 A-2 B \) b. \( 4 A+B \) c. \( A B \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
Mari kita hitung setiap bagian satu per satu.
### a. \( 3A - 2B \)
Pertama, kita hitung \( 3A \) dan \( 2B \).
\[
3A = 3 \left[\begin{array}{ccc} 2 & 1 & 3 \\ -4 & 5 & 2 \\ 3 & 4 & 1 \end{array}\right] = \left[\begin{array}{ccc} 6 & 3 & 9 \\ -12 & 15 & 6 \\ 9 & 12 & 3 \end{array}\right]
\]
\[
2B = 2 \left[\begin{array}{ccc} -5 & 2 & 3 \\ 2 & 4 & -5 \\ 1 & 3 & 4 \end{array}\right] = \left[\begin{array}{ccc} -10 & 4 & 6 \\ 4 & 8 & -10 \\ 2 & 6 & 8 \end{array}\right]
\]
Sekarang kita hitung \( 3A - 2B \):
\[
3A - 2B = \left[\begin{array}{ccc} 6 & 3 & 9 \\ -12 & 15 & 6 \\ 9 & 12 & 3 \end{array}\right] - \left[\begin{array}{ccc} -10 & 4 & 6 \\ 4 & 8 & -10 \\ 2 & 6 & 8 \end{array}\right]
\]
\[
= \left[\begin{array}{ccc} 6 - (-10) & 3 - 4 & 9 - 6 \\ -12 - 4 & 15 - 8 & 6 - (-10) \\ 9 - 2 & 12 - 6 & 3 - 8 \end{array}\right]
\]
\[
= \left[\begin{array}{ccc} 16 & -1 & 3 \\ -16 & 7 & 16 \\ 7 & 6 & -5 \end{array}\right]
\]
### b. \( 4A + B \)
Sekarang kita hitung \( 4A \) dan \( B \).
\[
4A = 4 \left[\begin{array}{ccc} 2 & 1 & 3 \\ -4 & 5 & 2 \\ 3 & 4 & 1 \end{array}\right] = \left[\begin{array}{ccc} 8 & 4 & 12 \\ -16 & 20 & 8 \\ 12 & 16 & 4 \end{array}\right]
\]
Sekarang kita hitung \( 4A + B \):
\[
4A + B = \left[\begin{array}{ccc} 8 & 4 & 12 \\ -16 & 20 & 8 \\ 12 & 16 & 4 \end{array}\right] + \left[\begin{array}{ccc} -5 & 2 & 3 \\ 2 & 4 & -5 \\ 1 & 3 & 4 \end{array}\right]
\]
\[
= \left[\begin{array}{ccc} 8 - 5 & 4 + 2 & 12 + 3 \\ -16 + 2 & 20 + 4 & 8 - 5 \\ 12 + 1 & 16 + 3 & 4 + 4 \end{array}\right]
\]
\[
= \left[\begin{array}{ccc} 3 & 6 & 15 \\ -14 & 24 & 3 \\ 13 & 19 & 8 \end{array}\right]
\]
### c. \( AB \)
Sekarang kita hitung hasil kali \( AB \).
\[
AB = \left[\begin{array}{ccc} 2 & 1 & 3 \\ -4 & 5 & 2 \\ 3 & 4 & 1 \end{array}\right] \left[\begin{array}{ccc} -5 & 2 & 3 \\ 2 & 4 & -5 \\ 1 & 3 & 4 \end{array}\right]
\]
Kita hitung setiap elemen dari hasil kali matriks:
- Elemen (1,1):
\[
2 \cdot (-5) + 1 \cdot 2 + 3 \cdot 1 = -10 + 2 + 3 = -5
\]
- Elemen (1,2):
\[
2 \cdot 2 + 1 \cdot 4 + 3 \cdot 3 = 4 + 4 + 9 = 17
\]
- Elemen (1,3):
\[
2 \cdot 3 + 1 \cdot (-5) + 3 \cdot 4 = 6 - 5 + 12 = 13
\]
- Elemen (2,1):
\[
-4 \cdot (-5) + 5 \cdot 2 + 2 \cdot 1 = 20 + 10 + 2 = 32
\]
- Elemen (2,2):
\[
-4 \cdot 2 + 5 \cdot 4 + 2 \cdot 3 = -8 + 20 + 6 = 18
\]
- Elemen (2,3):
\[
-4 \cdot 3 + 5 \cdot (-5) + 2 \cdot 4 = -12 - 25 + 8 = -29
\]
- Elemen (3,1):
\[
3 \cdot (-5) + 4 \cdot 2 + 1 \cdot 1 = -15 + 8 + 1 = -6
\]
- Elemen (3,2):
\[
3 \cdot 2 + 4 \cdot 4 + 1 \cdot 3 = 6 + 16 + 3 = 25
\]
- Elemen (3,3):
\[
3 \cdot 3 + 4 \cdot (-5) + 1 \cdot 4 = 9 - 20 + 4 = -7
\]
Jadi, hasil kali \( AB \) adalah:
\[
AB = \left[\begin{array}{ccc} -5 & 17 & 13 \\ 32 & 18 & -29 \\ -6 & 25 & -7 \end{array}\right]
\]
### Ringkasan Hasil
a. \( 3A - 2B = \left[\begin{array}{ccc} 16 & -1 & 3 \\ -16 & 7 & 16 \\ 7 & 6 & -5 \end{array}\right] \)
b. \( 4A + B = \left[\begin{array}{ccc} 3 & 6 & 15 \\ -14 & 24 & 3 \\ 13 & 19 & 8 \end{array}\right] \)
c. \( AB = \left[\begin{array}{ccc} -5 & 17 & 13 \\ 32 & 18 & -29 \\ -6 & 25 & -7 \end{array}\right] \)
Quick Answer
a. \( 3A - 2B = \left[\begin{array}{ccc} 16 & -1 & 3 \\ -16 & 7 & 16 \\ 7 & 6 & -5 \end{array}\right] \)
b. \( 4A + B = \left[\begin{array}{ccc} 3 & 6 & 15 \\ -14 & 24 & 3 \\ 13 & 19 & 8 \end{array}\right] \)
c. \( AB = \left[\begin{array}{ccc} -5 & 17 & 13 \\ 32 & 18 & -29 \\ -6 & 25 & -7 \end{array}\right] \)
Answered by UpStudy AI and reviewed by a Professional Tutor
UpStudy ThothAI
Self-Developed and Ever-Improving
Thoth AI product is constantly being upgraded and optimized.
Covers All Major Subjects
Capable of handling homework in math, chemistry, biology, physics, and more.
Instant and Accurate
Provides immediate and precise solutions and guidance.
Try Now
Ask Tutors
Ask AI
10x
Fastest way to Get Answers & Solutions
By text
Enter your question here…
By image
Re-Upload
Submit