When working with determinants, always ensure you're familiar with key properties like \( |\text{adj} \, A| = |A|^{n-1} \) and \( |A^{-1}| = \frac{1}{|A|} \). These properties are very helpful when simplifying matrix expressions and solving determinant-related problems. Be cautious with misstatements, such as \( |A| \neq |\text{adj} \, A|^{n-1} \), as they can lead to confusion.
For a square matrix \( A_{n \times n} \), the determinant of the adjugate of \( A \) is given by:
\[|\text{adj} \, A| = |A|^{n-1}.\]
This property confirms that (A) is correct.
For the inverse of a matrix:
\[|A^{-1}| = \frac{1}{|A|}.\]
This property confirms that (D) is correct.
(C) is not part of the correct answer because while the relation \( A (\text{adj} \, A) = |A| I \) is valid, it is not relevant to the determinant properties discussed here.
(B) is incorrect because \( |A| \neq |\text{adj} \, A|^{n-1} \). It is a misstatement of the property.
Thus, the correct options are:
\((A)\) and \((D)\).
For a square matrix \( A_{n \times n} \), the determinant of the adjugate of \( A \) is given by:
\[ |\text{adj} \, A| = |A|^{n-1}. \]
This property confirms that (A) is correct.
For the inverse of a matrix:
\[ |A^{-1}| = \frac{1}{|A|}. \]
This property confirms that (D) is correct.
Explanation for (C):
While the relation \( A (\text{adj} \, A) = |A| I \) is valid, it is not relevant to the determinant properties discussed here. Hence, (C) is not part of the correct answer.
Explanation for (B):
(B) is incorrect because \( |A| \neq |\text{adj} \, A|^{n-1} \). It is a misstatement of the property.
Conclusion:
The correct options are:
List-I | List-II | ||
A | Megaliths | (I) | Decipherment of Brahmi and Kharoshti |
B | James Princep | (II) | Emerged in first millennium BCE |
C | Piyadassi | (III) | Means pleasant to behold |
D | Epigraphy | (IV) | Study of inscriptions |