Matrices & Determinants
Matrices & Determinants
star_batch_jee_advanced_2025
Grade None
Question:
$A_1$ is a matrix formed by replacing all the elements in $A_{n \times n}$ by corresponding cofactors. $A_2$ is matrix formed by replacing all elements of $A_1$ by corresponding cofactors and $A_3, A_4 \ldots$ formed so on. if $|A_n| = K$, then $|A| =$
K^{n^2}
K^{(n-1)^2}
K^{(n-1)^n}
\frac{1}{K^{(n-1)^n}}
Step-by-Step Solution
Key Concept: The determinant of the adjugate (cofactor transpose) of an $n \times n$ matrix $A$ is $|A|^{n-1}$.
If $A_1$ is the cofactor matrix of $A$, then $A_1 = (\text{Adj}(A))^T$ and $|A_1| = |\text{Adj}(A)| = |A|^{n-1}$. For $A_2$ as the cofactor matrix of $A_1$, we have $|A_2| = |A_1|^{n-1} = (|A|^{n-1})^{n-1} = |A|^{(n-1)^2}$. Continuing this pattern, $|A_n| = |A|^{(n-1)^n}$.
Correct Answer: 3