Math Problem Statement
Let A = [[1, 2], [3, 6]]. Find a 2x3 matrix B with distinct nonzero entries such that AB = 0.
Solution
We are given the matrix as:
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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Linear Algebra
Null Space
Formulas
Matrix Multiplication: (A * B) = C
Linear System Solution: A * X = 0
Theorems
Rank-Nullity Theorem
Suitable Grade Level
Undergraduate Level (Linear Algebra Course)
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