matrix
3. Tentukan matriks X!
4. Tentukan A×B dan 2A×B!
5. Tentukan nilai a dan b yang memenuhi persamaan matriks!
3. Matriks X adalah [tex]\left(\begin{array}{cc}- 1&5\\4&- 14\end{array}\right)[/tex].
4. Nilai A×B adalah [tex]\left(\begin{array}{cc}41&15\\52&22\end{array}\right)[/tex].
Nilai 2A×B adalah [tex]\left(\begin{array}{cc}82&30\\104&64\end{array}\right)[/tex]
5. Nilai a yang memenuhi persamaan adalah 2,8.
Nilai b yang memenuhi persamaan adalah 5.
Penjelasan dengan langkah-langkah:
Diketahui:
- 3. [tex]\left(\begin{array}{cc}- 5&7\\4&10\end{array}\right) \:+\: X \:=\: \left(\begin{array}{cc}- 6&12\\8&- 4\end{array}\right)[/tex]
- 4. A = [tex]\left(\begin{array}{ccc}10&- 1&5\\- 2&3&6\end{array}\right)[/tex]
B = [tex]\left(\begin{array}{cc}1&- 1\\4&0\\7&5\end{array}\right)[/tex] - 5. [tex]\left(\begin{array}{cc}a&- 2\\3&b\end{array}\right) \times \left(\begin{array}{cc}- 3\\a\end{array}\right) \:=\: \left(\begin{array}{cc}- 14\\5\end{array}\right)[/tex]
Ditanyakan:
- 3. X?
- 4. A×B?
2A×B? - 5. a? b?
Jawaban:
3. [tex]\left(\begin{array}{cc}- 5&7\\4&10\end{array}\right) \:+\: X \:=\: \left(\begin{array}{cc}- 6&12\\8&- 4\end{array}\right)[/tex]
[tex]X \:=\: \left(\begin{array}{cc}- 6&12\\8&- 4\end{array}\right) \:-\: \left(\begin{array}{cc}- 5&7\\4&10\end{array}\right)[/tex]
[tex]X \:=\: \left(\begin{array}{cc}- 6 \:+\: 5&12 \:-\: 7\\8 \:-\: 4&- 4 \:-\: 10\end{array}\right)[/tex]
[tex]X \:=\: \left(\begin{array}{cc}- 1&5\\4&- 14\end{array}\right)[/tex]
4. A×B = [tex]\left(\begin{array}{ccc}10&- 1&5\\- 2&3&6\end{array}\right) \times \left(\begin{array}{cc}1&- 1\\4&0\\7&5\end{array}\right)[/tex]
A×B = [tex]\left(\begin{array}{cc}(10 \times 1 \:+\: - 1 \times 4 \:+\: 5 \times 7)&(10 \times - 1 \:+\: - 1 \times 0 \:+\: 5 \times 5)\\(- 2 \times 1 \:+\: 3 \times 4 \:+\: 6 \times 7)&(- 2 \times - 1 \:+\: 3 \times 0 \:+\: 6 \times 5)\end{array}\right)[/tex]
A×B = [tex]\left(\begin{array}{cc}(10 \:-\: 4 \:+\: 35)&(- 10 \:+\: 0 \:+\: 25)\\(- 2 \:+\: 12 \:+\: 42)&(2 \:+\: 0 \:+\: 20)\end{array}\right)[/tex]
A×B = [tex]\left(\begin{array}{cc}41&15\\52&22\end{array}\right)[/tex]
Menentukan nilai 2A.
2A = [tex]2 \times A[/tex]
2A = [tex]2 \times \left(\begin{array}{ccc}10&- 1&5\\- 2&3&6\end{array}\right)[/tex]
2A = [tex]\left(\begin{array}{ccc}20&- 2&10\\- 4&6&12\end{array}\right)[/tex]
Mengalikan 2A dengan B.
2A×B = [tex]\left(\begin{array}{ccc}20&- 2&10\\- 4&6&12\end{array}\right) \times \left(\begin{array}{cc}1&- 1\\4&0\\7&5\end{array}\right)[/tex]
2A×B = [tex]\left(\begin{array}{cc}(20 \times 1 \:+\: - 2 \times 4 \:+\: 10 \times 7)&(20 \times - 1 \:+\: - 2 \times 0 \:+\: 10 \times 5)\\(- 4 \times 1 \:+\: 6 \times 4 \:+\: 12 \times 7)&(- 4 \times - 1 \:+\: 6 \times 0 \:+\: 12 \times 5)\end{array}\right)[/tex]
2A×B = [tex]\left(\begin{array}{cc}(20 \:-\: 8 \:+\: 70)&(- 20 \:+\: 0 \:+\: 50)\\(- 4 \:+\: 24 \:+\: 84)&(4 \:+\: 0 \:+\: 60)\end{array}\right)[/tex]
2A×B = [tex]\left(\begin{array}{cc}82&30\\104&64\end{array}\right)[/tex]
5. [tex]\left(\begin{array}{cc}a&- 2\\3&b\end{array}\right) \times \left(\begin{array}{cc}- 3\\a\end{array}\right) \:=\: \left(\begin{array}{cc}- 14\\5\end{array}\right)[/tex]
[tex]\left(\begin{array}{cc}a \times - 3 \:+\: - 2 \times a\\3 \times - 3 \:+\: b \times a\end{array}\right) \:=\: \left(\begin{array}{cc}- 14\\5\end{array}\right)[/tex]
[tex]\left(\begin{array}{cc}- 3a \:+\: - 2a\\- 9 \:+\: ab\end{array}\right) \:=\: \left(\begin{array}{cc}- 14\\5\end{array}\right)[/tex]
[tex]\left(\begin{array}{cc}- 5a\\- 9 \:+\: ab\end{array}\right) \:=\: \left(\begin{array}{cc}- 14\\5\end{array}\right)[/tex]
Maka
- - 5a = - 14
a = [tex]\frac{- 14}{- 5}[/tex]
a = 2,8 - - 9 + ab = 5
ab = 5 + 9
ab = 14
b = [tex]14 \div a[/tex]
b = [tex]14 \div \frac{14}{5}[/tex]
b = [tex]14 \times \frac{5}{14}[/tex]
b = 5
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