Jika AC = B dan C^-1 invers matriks C, determinan dari matriks C^-1 adalah

Diketahui matriks A = [tex]\left[\begin{array}{cc}4&2\\3&-4\end{array}\right][/tex] dan matriks B = [tex]\left[\begin{array}{cc}5&-3\\2&1\end{array}\right][/tex]. Jika AC = B dan C⁻¹ invers matriks C, determinan dari matriks C⁻¹ adalah ...

A. -2            D. 2

B. -1             E. 3

C. 1

Jawaban

Pendahuluan  

Jika A =  [tex]\left[\begin{array}{cc}a&b\\c&d\end{array}\right][/tex]

Determinan A = |A| = ad – bc

Invers matriks A = A⁻¹ =  [tex]\frac{1}{|A|}\left[\begin{array}{cc}d&-b\\-c&a\end{array}\right][/tex]

Sifat determinan matriks

1. |A . B| = |A| . |B|

2. |A⁻¹| =  

3. |Aᵗ| = |A|

Sifat Invers matriks

1. (A⁻¹)⁻¹ = A

2. (AB)⁻¹ = B⁻¹ . A⁻¹

3. A . A⁻¹ = A⁻¹ . A = I, dengan I = matriks identitas  

4. AX = B ⇒ X = A⁻¹B

5. XA = B ⇒ X = BA⁻¹

Pembahasan  

A =  [tex]\left[\begin{array}{cc}4&2\\3&-4\end{array}\right][/tex]

|A| = 4(– 4) – 2(3) =  – 16 – 6 =  – 22

B =  [tex]\left[\begin{array}{cc}5&-3\\2&1\end{array}\right][/tex]

|B| = 5(1) – (–3)(2) = 5 + 6 = 11

Cara 1

AC = B

C = A⁻¹.B

C⁻¹ = (A⁻¹.B)⁻¹

C⁻¹ = B⁻¹ . A

[tex]C^{-1}=\frac{1}{11} \left[\begin{array}{cc}1&3\\-2&5\end{array}\right]\left[\begin{array}{cc}4&2\\3&-4\end{array}\right]\\ \\ C^{-1}=\frac{1}{11} \left[\begin{array}{cc}4+9&2-12\\-8+15&-4-20\end{array}\right]\\ \\ C^{-1}=\frac{1}{11} \left[\begin{array}{cc}13&-10\\7&-24\end{array}\right]\\ \\ C^{-1}= \left[\begin{array}{cc}\frac{13}{11}&-\frac{10}{11}\\\frac{7}{11}&-\frac{24}{11}\end{array}\right][/tex]

[tex]|C^{-1}|=\frac{13}{11} .(-\frac{24}{11})-(-\frac{10}{11}).\frac{7}{11}\\ \\ |C^{-1}|=-\frac{312}{121} +\frac{70}{121}\\ \\ |C^{-1}|=-\frac{242}{121}\\ \\ |C^{-1}|=-2[/tex]

Cara 2

AC = B

|AC| = |B|

|A| . |C| = |B|

–22 . |C| = 11

|C| =  [tex]\frac{11}{-22} =-\frac{1}{2}[/tex]

|C⁻¹| = [tex]-\frac{2}{1}[/tex] = –2

Jawaban A

Kesimpulan

Determinan dari matriks C⁻¹ adalah –2

Pelajari lebih lanjut    

https://brainly.co.id/tugas/12724799

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Detil Jawaban  

Kelas : 11

Mapel : Matematika  

Kategori : Matriks

Kode : 11.2.5  

Kata Kunci : determinan matriks invers