Nichols Goodwin
05/07/2023 · Junior High School
Jika \( A=\left[\begin{array}{lll}1 & 2 & 0 \\ 3 & 4 & 2\end{array}\right] \) dan \( B=\left[\begin{array}{ll}4 & 2 \\ -1 & 1 \\ 0 & 0\end{array}\right] \operatorname{maka}(B A)^{T}=\ldots \ldots \ldots \ldots \)
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Step-by-step Solution
Untuk menghitung \((BA)^T\), kita perlu terlebih dahulu menghitung hasil kali matriks \(BA\).
Matriks \(A\) berukuran \(2 \times 3\) dan matriks \(B\) berukuran \(3 \times 2\). Hasil kali \(BA\) akan menjadi matriks berukuran \(3 \times 3\).
Mari kita hitung \(BA\):
\[
B = \begin{bmatrix}
4 & 2 \\
-1 & 1 \\
0 & 0
\end{bmatrix}, \quad A = \begin{bmatrix}
1 & 2 & 0 \\
3 & 4 & 2
\end{bmatrix}
\]
Hasil kali \(BA\) dapat dihitung sebagai berikut:
\[
BA = \begin{bmatrix}
4 & 2 \\
-1 & 1 \\
0 & 0
\end{bmatrix} \begin{bmatrix}
1 & 2 & 0 \\
3 & 4 & 2
\end{bmatrix}
\]
Mari kita hitung setiap elemen dari matriks \(BA\):
1. Elemen (1,1):
\[
4 \cdot 1 + 2 \cdot 3 = 4 + 6 = 10
\]
2. Elemen (1,2):
\[
4 \cdot 2 + 2 \cdot 4 = 8 + 8 = 16
\]
3. Elemen (1,3):
\[
4 \cdot 0 + 2 \cdot 2 = 0 + 4 = 4
\]
4. Elemen (2,1):
\[
-1 \cdot 1 + 1 \cdot 3 = -1 + 3 = 2
\]
5. Elemen (2,2):
\[
-1 \cdot 2 + 1 \cdot 4 = -2 + 4 = 2
\]
6. Elemen (2,3):
\[
-1 \cdot 0 + 1 \cdot 2 = 0 + 2 = 2
\]
7. Elemen (3,1):
\[
0 \cdot 1 + 0 \cdot 3 = 0 + 0 = 0
\]
8. Elemen (3,2):
\[
0 \cdot 2 + 0 \cdot 4 = 0 + 0 = 0
\]
9. Elemen (3,3):
\[
0 \cdot 0 + 0 \cdot 2 = 0 + 0 = 0
\]
Sehingga, hasil kali \(BA\) adalah:
\[
BA = \begin{bmatrix}
10 & 16 & 4 \\
2 & 2 & 2 \\
0 & 0 & 0
\end{bmatrix}
\]
Selanjutnya, kita perlu menghitung transpos dari matriks \(BA\):
\[
(BA)^T = \begin{bmatrix}
10 & 2 & 0 \\
16 & 2 & 0 \\
4 & 2 & 0
\end{bmatrix}
\]
Jadi, hasil akhir dari \((BA)^T\) adalah:
\[
(BA)^T = \begin{bmatrix}
10 & 2 & 0 \\
16 & 2 & 0 \\
4 & 2 & 0
\end{bmatrix}
\]
Quick Answer
\[
(BA)^T = \begin{bmatrix}
10 & 2 & 0 \\
16 & 2 & 0 \\
4 & 2 & 0
\end{bmatrix}
\]
Answered by UpStudy AI and reviewed by a Professional Tutor
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