Question

Let \( A = \begin{pmatrix} 0 & 3 & 5 & -7 \\ 8 & 0 & -1 & 0 \\ 6 & -1 & 0 & 0 \end{pmatrix} \) and \( X = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \). If \( D = [\alpha \, \beta \, \gamma]^T \) is the solution of \( X^T B^T = A^T X \), then \( D^T A = \)

• A. \( 0 \)
• B. \( 4 \)
• C. \( -2 \)
• D. \( 6 \)

Hint:

When dealing with matrix equations, always carefully use the properties of matrix transposition and multiplication to simplify.

Answer 1

The Correct Option is B

We are given matrices \( A \), \( B \), and the equation \( X^T B^T = A^T X \). To find \( D^T A \), we can use the properties of matrices.
Step 1: First, compute the transpose of matrix \( A \), denoted as \( A^T \), and the matrix \( B^T \).
Step 2: Use the equation \( X^T B^T = A^T X \) to find \( D \), and calculate \( D^T A \).
After solving, the result is \( 4 \), which is the correct answer.