Matrix Algebra

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

Option (1): For any invertible matrix , the adjugate matrix is well-defined and non-singular. Hence, its determinant is non-zero:
Option (2): An invertible matrix by definition has a non-zero determinant:
Option (3): The product , and the determinant of the identity matrix is:
Option (4): Recall the identity: Taking determinants on both sides (for an matrix): Therefore:
Conclusion:
Since Option (4) incorrectly implies , which is not true for general invertible matrices, it is the false statement.

✅ Correct Answer: Option (4)