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We are given the matrix \( A \) as:

\[
A = \begin{bmatrix} 
1 & 2 \\
3 & 6 
\end{bmatrix}
\]

We need to find a \( 2 \times 3 \) matrix \( B \), with distinct nonzero entries, such that \( AB = 0 \). This means we want:

\[
A \times B = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}
\]

Let's denote the matrix \( B \) as:

\[
B = \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{bmatrix}
\]

The product \( AB \) is computed as follows:

\[
AB = \begin{bmatrix} 
1 & 2 \\
3 & 6 
\end{bmatrix}
\times
\begin{bmatrix} 
b_{11} & b_{12} & b_{13} \\
b_{21} & b_{22} & b_{23}
\end{bmatrix}
\]

This results in:

\[
AB = \begin{bmatrix} 
1 \cdot b_{11} + 2 \cdot b_{21} & 1 \cdot b_{12} + 2 \cdot b_{22} & 1 \cdot b_{13} + 2 \cdot b_{23} \\
3 \cdot b_{11} + 6 \cdot b_{21} & 3 \cdot b_{12} + 6 \cdot b_{22} & 3 \cdot b_{13} + 6 \cdot b_{23}
\end{bmatrix}
=
\begin{bmatrix} 
0 & 0 & 0 \\
0 & 0 & 0 
\end{bmatrix}
\]

From this, we obtain the system of equations:

1. \( 1 \cdot b_{11} + 2 \cdot b_{21} = 0 \)
2. \( 1 \cdot b_{12} + 2 \cdot b_{22} = 0 \)
3. \( 1 \cdot b_{13} + 2 \cdot b_{23} = 0 \)
4. \( 3 \cdot b_{11} + 6 \cdot b_{21} = 0 \)
5. \( 3 \cdot b_{12} + 6 \cdot b_{22} = 0 \)
6. \( 3 \cdot b_{13} + 6 \cdot b_{23} = 0 \)

Notice that equations 4, 5, and 6 are just multiples of equations 1, 2, and 3, respectively, and hence do not provide any new information. Therefore, we are left with:

1. \( b_{11} + 2b_{21} = 0 \)
2. \( b_{12} + 2b_{22} = 0 \)
3. \( b_{13} + 2b_{23} = 0 \)

We can solve for \( b_{11}, b_{12}, b_{13} \) in terms of \( b_{21}, b_{22}, b_{23} \):

\[
b_{11} = -2b_{21}, \quad b_{12} = -2b_{22}, \quad b_{13} = -2b_{23}
\]

Now, we just need to assign distinct, nonzero values for \( b_{21}, b_{22}, b_{23} \). For example, we can choose:

\[
b_{21} = 1, \quad b_{22} = 2, \quad b_{23} = 3
\]

This gives:

\[
b_{11} = -2, \quad b_{12} = -4, \quad b_{13} = -6
\]

Thus, the matrix \( B \) is:

\[
B = \begin{bmatrix} 
-2 & -4 & -6 \\
1 & 2 & 3 
\end{bmatrix}
\]

This matrix satisfies the condition \( AB = 0 \) and has distinct, nonzero entries.

---

Would you like more details on any part of this solution? Here are some related questions for practice:

1. How would you check if a matrix product results in a zero matrix?
2. What is the general form of a matrix equation \( AX = 0 \)?
3. Can you generalize this method to find a \( 3 \times 2 \) matrix with a similar property?
4. How would you verify if the matrix \( B \) is unique for this problem?
5. What is the rank of matrix \( A \) in this problem?

**Tip:** When dealing with matrix multiplication, pay attention to the dimensions of the matrices involved to ensure the operations are valid.

Let A = [[1, 2], [3, 6]]. Find a 2x3 matrix B with distinct nonzero entries such that AB = 0.

This problem focuses on finding a 2x3 matrix B with distinct nonzero entries such that multiplying matrix A by B results in the zero matrix. It covers concepts like matrix multiplication, linear systems, and the null space.

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