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: Understanding the Given Determinants
Given that \( |B| = 5 \) and using the determinant property: \[ |BA^T| = |B| \cdot |A^T| \] Since \( |A^T| = |A| \), we get: \[ |B| \cdot |A| = 15 \] Substituting \( |B| = 5 \), we solve for \( |A| \): \[ 5 \cdot |A| = 15 \Rightarrow |A| = 3 \]
Step 2: Using the Determinant Property for \( P^T AP \) \[ |P^T AP| = |P^T| \cdot |A| \cdot |P| \] Since \( |P^T| = |P| \), we simplify: \[ |P|^2 \cdot |A| = -27 \] Substituting \( |A| = 3 \): \[ |P|^2 \cdot 3 = -27 \] \[ |P|^2 = 9 \] \[ |P| = \pm 3 \]
Step 3: Matching the Answer Options
The correct answer from the given choices is \( |P| = 3 \). Final Answer: (a) 3.