Question

Let \( A \) be a \( 3 \times 3 \) real matrix such that \[ A \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 4 \\ 0 \end{bmatrix}, \quad A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 4 \\ 0 \\ 0 \end{bmatrix}, \quad A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 4 \end{bmatrix}. \] Then which of the following statements is/are true?

• A. \( A \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 2 \\ 2 \\ -2 \end{bmatrix} \)
• B. \( A \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 2 \\ -2 \\ 2 \end{bmatrix} \)
• C. \( A \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 2 \\ 0 \\ 2 \end{bmatrix} \)
• D. \( A \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} = \begin{bmatrix} 8 \\ 4 \\ 0 \end{bmatrix} \)

Hint:

- To solve matrix-related problems, first find the matrix elements using the given vectors and their corresponding outcomes.
- Verify each option by performing the matrix multiplication and comparing the results.

Answer 1

The Correct Option is A

The given problem provides the results of multiplying matrix \( A \) with specific vectors. Based on this information, we can deduce the matrix \( A \). By using the provided equations, we construct a system to find the matrix elements. \[ A = \begin{bmatrix} 2 & 2 & -2 \\ 2 & -2 & 2 \\ 8 & 4 & 0 \end{bmatrix} \] Now, checking the options:
(A) We check if multiplying \( A \) with \( \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \) gives \( \begin{bmatrix} 2 \\ 2 \\ -2 \end{bmatrix} \), which it does.
Thus, the correct answer is (A).