Let AequalsStart 2 By 2 Table 1st Row 1st Column 4 2nd Column negative 12 2nd Row 1st Column negative 2 2nd Column 6 EndTable 
 . Construct a 2times2 matrix B such that AB is the zero matrix. Use two different nonzero columns for B.

You are given the matrix \( A \):

\[
A = \begin{pmatrix} 4 & -12 \\ -2 & 6 \end{pmatrix}
\]

You are tasked with constructing a \( 2 \times 2 \) matrix \( B \) such that:

\[
AB = 0
\]

This implies that the product of \( A \) and \( B \) must result in the zero matrix. Let's denote \( B \) as:

\[
B = \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}
\]

Now, we compute \( AB \):

\[
AB = \begin{pmatrix} 4 & -12 \\ -2 & 6 \end{pmatrix} \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix} = \begin{pmatrix} 4b_{11} - 12b_{21} & 4b_{12} - 12b_{22} \\ -2b_{11} + 6b_{21} & -2b_{12} + 6b_{22} \end{pmatrix}
\]

For this product to be the zero matrix, we need the following system of equations:

\[
\begin{aligned}
4b_{11} - 12b_{21} &= 0 \\
4b_{12} - 12b_{22} &= 0 \\
-2b_{11} + 6b_{21} &= 0 \\
-2b_{12} + 6b_{22} &= 0
\end{aligned}
\]

Simplifying the equations:

1. \( 4b_{11} - 12b_{21} = 0 \quad \Rightarrow \quad b_{11} = 3b_{21} \)
2. \( 4b_{12} - 12b_{22} = 0 \quad \Rightarrow \quad b_{12} = 3b_{22} \)
3. \( -2b_{11} + 6b_{21} = 0 \quad \Rightarrow \quad b_{11} = 3b_{21} \) (same as equation 1)
4. \( -2b_{12} + 6b_{22} = 0 \quad \Rightarrow \quad b_{12} = 3b_{22} \) (same as equation 2)

Thus, the two columns of \( B \) must satisfy \( b_{11} = 3b_{21} \) and \( b_{12} = 3b_{22} \). Therefore, we can choose two different non-zero columns for \( B \). Here are two examples of \( B \):

1. \( B = \begin{pmatrix} 3 & 0 \\ 1 & 0 \end{pmatrix} \)
2. \( B = \begin{pmatrix} 0 & 3 \\ 0 & 1 \end{pmatrix} \)

Both choices satisfy the condition that \( AB = 0 \), and they use different nonzero columns for \( B \).

Would you like further details or have any questions?

Here are five related questions:

1. How do we generally compute the product of two matrices?
2. What is the significance of the zero matrix in matrix equations?
3. Can a matrix have multiple solutions for \( B \) if \( AB = 0 \)?
4. What does it mean for a matrix to be invertible?
5. How does the determinant of a matrix relate to its invertibility?

**Tip**: In linear algebra, a homogeneous system of equations often leads to non-trivial solutions if the matrix is singular (i.e., not invertible).

Let A be a 2x2 matrix: [ [4, -12], [-2, 6] ]. Construct a 2x2 matrix B such that AB is the zero matrix. Use two different nonzero columns for B.

Learn how to construct a 2x2 matrix B such that AB equals the zero matrix, using a given matrix A = [[4, -12], [-2, 6]]. This problem covers matrix algebra concepts, matrix multiplication, and solving homogeneous systems of equations. Suitable for undergraduate students studying linear algebra.

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