Question

A, P, B are \( 3 \times 3 \) matrices. If \( |B| = 5 \), \( | BA^T | = 15 \), \( | P^T AP | = -27 \), then one of the values of \( | P | \) is:

• A. 3
• B. -5
• C. 9
• D. 6

Hint:

When solving determinant equations involving matrix properties, always use: \[ |AB| = |A||B|, \quad |A^T| = |A| \]

Answer 1

The Correct Option is A

Step 1: Determine \( |A| \)
Given \( |B| = 5 \). Using the property \( |XY| = |X||Y| \) and \( |A^T| = |A| \), we have \( |BA^T| = |B||A^T| = |B||A| \).
Thus, \( |B||A| = 15 \).
Substituting \( |B| = 5 \), we get \( 5|A| = 15 \), which implies \( |A| = 3 \).

Step 2: Determine \( |P| \)
Using the property \( |XY| = |X||Y| \) and \( |P^T| = |P| \), we have \( |P^T AP| = |P^T||A||P| = |P|^2|A| \).
Therefore, \( |P|^2|A| = -27 \).
Substituting \( |A| = 3 \), we get \( |P|^2 \cdot 3 = -27 \).
This simplifies to \( |P|^2 = -9 \).
The problem statement contains an error as the square of a real determinant cannot be negative. However, proceeding with the calculation as if \( |P|^2 = 9 \) (assuming a typo in the original problem), we find \( |P| = \pm 3 \).

Step 3: Select the Answer
Based on the provided answer options, the correct value for \( |P| \) is \( 3 \).

Final Answer: (a) 3.