Question

If \( A = \begin{pmatrix} -7 & -4 \\ 7 & 4 \end{pmatrix} \) and \( B = \begin{pmatrix} 4 & 1 \\ 2 & 7 \end{pmatrix} \), then which statement is true?

• A. \( A^T A = I \)
• B. \( B^T = I \)
• C. \( AB = BA \)
• D. \( (AB)^T = I \)

Hint:

To verify properties of matrices, use matrix multiplication and properties such as transposition and the identity matrix.

Answer 1

The Correct Option is A

Step 1: Check if \( A^T A = I \), where \( A^T \) is the transpose of \( A \) and \( I \) is the identity matrix.
Step 2: Calculate the transpose and product of \( A^T A \), which is found to be the identity matrix.

Final Answer: \[ \boxed{A^T A = I} \]