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We are given two matrices:

Matrix \( A = \begin{pmatrix} 5 \\ 1 \\ 3 \end{pmatrix} \) (a \(3 \times 1\) column vector), and  
Matrix \( B = \begin{pmatrix} 4 & 1 & 0 \\ 0 & 2 \end{pmatrix} \) (a \( 2 \times 3 \) matrix).

The question asks to determine the result of various operations between \( A \) and \( B \). These operations involve matrix multiplication, as well as the transposes of these matrices.

Let's break down each option:

### a) \( A \times B^t \) 
This means multiplying matrix \( A \) with the transpose of \( B \), \( B^t \).

\( B^t = \begin{pmatrix} 4 & 0 \\ 1 & 2 \\ 0 & 2 \end{pmatrix} \) (a \( 3 \times 2 \) matrix).

The multiplication \( A \times B^t \) is not defined because \( A \) is a \( 3 \times 1 \) matrix and \( B^t \) is a \( 3 \times 2 \) matrix. For multiplication to be valid, the number of columns in the first matrix must match the number of rows in the second matrix, which is not the case here.

### b) \( B \times A^t \)
Here, \( A^t = \begin{pmatrix} 5 & 1 & 3 \end{pmatrix} \) (a \( 1 \times 3 \) matrix). Matrix \( B \) is \( 2 \times 3 \), so the multiplication \( B \times A^t \) is valid.

Perform the multiplication:
\[
B \times A^t = \begin{pmatrix} 4 & 1 & 0 \\ 0 & 2 & 2 \end{pmatrix} \times \begin{pmatrix} 5 \\ 1 \\ 3 \end{pmatrix}
\]
\[
= \begin{pmatrix} 4(5) + 1(1) + 0(3) \\ 0(5) + 2(1) + 2(3) \end{pmatrix}
= \begin{pmatrix} 20 + 1 + 0 \\ 0 + 2 + 6 \end{pmatrix}
= \begin{pmatrix} 21 \\ 8 \end{pmatrix}
\]
So, the result of \( B \times A^t \) is \( \begin{pmatrix} 21 \\ 8 \end{pmatrix} \).

### c) \( A \times B \)
This multiplication is not defined. \( A \) is \( 3 \times 1 \), and \( B \) is \( 2 \times 3 \). The number of columns in \( A \) does not match the number of rows in \( B \).

### d) \( A^t \times B \)
Matrix \( A^t \) is \( 1 \times 3 \), and \( B \) is \( 2 \times 3 \). The multiplication \( A^t \times B \) is not valid for the same reason as in part (a).

### e) \( B^t \times A \)
Here, \( B^t \) is a \( 3 \times 2 \) matrix, and \( A \) is a \( 3 \times 1 \) matrix. The multiplication \( B^t \times A \) is valid.

Perform the multiplication:
\[
B^t \times A = \begin{pmatrix} 4 & 0 \\ 1 & 2 \\ 0 & 2 \end{pmatrix} \times \begin{pmatrix} 5 \\ 1 \\ 3 \end{pmatrix}
\]
However, this calculation should be re-evaluated because there's a conflict in matrix size requirements.

The final

Jadi A = [5, 1, 3] dan B = [4, 1, 0, 2], tentukan hasil dari operasi yang diberikan.

Learn how to solve a matrix multiplication problem involving matrices A and B, including transpose operations and matrix multiplication. Detailed step-by-step solution included. Suitable for high school students (Grades 10-12).

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