Chadwick Vaughan
11/28/2023 · Middle School
Manakah dari matriks berikut yang termasuk jenis matriks singular: \begin{tabular}{ll} a. \( \left(\begin{array}{cc}2 & 3 \\ 3 & -5\end{array}\right) \) & c. \( _{1}\left(\begin{array}{cc}\sqrt{3} & 3 \\ -\sqrt{2} & -\sqrt{6}\end{array}\right) \) \\ b. \( \left(\begin{array}{cc}1 & -2 \\ -1 & 2\end{array}\right) \) & d. \( _{1}\left(\begin{array}{cc}\sin ^{2} x & 2 \\ -0,5 & \cos ^{2} x\end{array}\right) \) \\ \hline\end{tabular}
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Tutor-Verified Answer
Step-by-step Solution
Matriks singular adalah matriks yang tidak memiliki invers, yang berarti determinannya sama dengan nol. Mari kita hitung determinan dari masing-masing matriks yang diberikan.
a. Matriks \( \left(\begin{array}{cc}2 & 3 \\ 3 & -5\end{array}\right) \):
\[
\text{Determinant} = (2)(-5) - (3)(3) = -10 - 9 = -19 \quad (\text{tidak singular})
\]
b. Matriks \( \left(\begin{array}{cc}1 & -2 \\ -1 & 2\end{array}\right) \):
\[
\text{Determinant} = (1)(2) - (-2)(-1) = 2 - 2 = 0 \quad (\text{singular})
\]
c. Matriks \( \left(\begin{array}{cc}\sqrt{3} & 3 \\ -\sqrt{2} & -\sqrt{6}\end{array}\right) \):
\[
\text{Determinant} = (\sqrt{3})(-\sqrt{6}) - (3)(-\sqrt{2}) = -\sqrt{18} + 3\sqrt{2}
\]
Kita perlu menghitung nilai ini lebih lanjut, tetapi kita bisa lihat bahwa ini tidak langsung menghasilkan 0. Mari kita hitung:
\[
-\sqrt{18} + 3\sqrt{2} = -3\sqrt{2} + 3\sqrt{2} = 0 \quad (\text{singular})
\]
d. Matriks \( \left(\begin{array}{cc}\sin ^{2} x & 2 \\ -0.5 & \cos ^{2} x\end{array}\right) \):
\[
\text{Determinant} = (\sin^2 x)(\cos^2 x) - (2)(-0.5) = \sin^2 x \cos^2 x + 1
\]
Karena \(\sin^2 x \cos^2 x\) selalu non-negatif, maka determinan ini tidak akan pernah sama dengan 0. Jadi, matriks ini tidak singular.
Kesimpulannya, matriks yang termasuk jenis matriks singular adalah:
- b. \( \left(\begin{array}{cc}1 & -2 \\ -1 & 2\end{array}\right) \)
- c. \( \left(\begin{array}{cc}\sqrt{3} & 3 \\ -\sqrt{2} & -\sqrt{6}\end{array}\right) \)
Quick Answer
b. \( \left(\begin{array}{cc}1 & -2 \\ -1 & 2\end{array}\right) \)
c. \( \left(\begin{array}{cc}\sqrt{3} & 3 \\ -\sqrt{2} & -\sqrt{6}\end{array}\right) \)
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