Question

For an invertible matrix \( A \), which of the following is not always true:

• A. \( \left| {adj}(A) \right| \neq 0 \)
• B. \( \left| A \right| \neq 0 \)
• C. \( \left| AA^{-1} \right| = 1 \)
• D. \( \left| A \, {adj}(A) \right| \neq 1 \)

Hint:

For an invertible matrix \( A \), the adjugate matrix \( {adj}(A) \) satisfies \( A \, {adj}(A) = \left| A \right| I \), where \( I \) is the identity matrix.

Answer 1

The Correct Option is D

We are given that \( A \) is an invertible matrix. Let's analyze the four options:

Option (1): For any invertible matrix \( A \), the adjugate matrix \( \text{adj}(A) \) is well-defined and non-singular. Hence, its determinant is non-zero: \[ \left| \text{adj}(A) \right| \neq 0 \]
Option (2): An invertible matrix by definition has a non-zero determinant: \[ \left| A \right| \neq 0 \]
Option (3): The product \( AA^{-1} = I \), and the determinant of the identity matrix is: \[ \left| AA^{-1} \right| = \left| I \right| = 1 \]
Option (4): Recall the identity: \[ A \cdot \text{adj}(A) = |A| \cdot I \] Taking determinants on both sides (for an \( n \times n \) matrix): \[ \left| A \cdot \text{adj}(A) \right| = \left| |A| \cdot I \right| = |A|^n \] Therefore: \[ \left| A \cdot \text{adj}(A) \right| = |A|^n, \quad \text{not } 1 \text{ in general} \]
Conclusion:
Since Option (4) incorrectly implies \( \left| A \cdot \text{adj}(A) \right| = 1 \), which is not true for general invertible matrices, it is the false statement.

✅ Correct Answer: Option (4)