Match List-I with List-II

List-I	List-II
(A) If $A$ is a non-singular matrix of order $n$, then $\vert A(\text{adj} A)\vert$ is equal to	(I) $\vert A\vert ^{n-1}$
(B) If $A$ is a non-singular matrix of order $n$, then $\vert \text{adj}(\text{adj} A)\vert$ is equal to	(II) $\vert A\vert ^{n-2} A$
(C) If $A$ is a non-singular matrix of order $n$, then $\text{adj}(\text{adj} A)$ is equal to	(III) $\vert A\vert ^n$
(D) If $A$ is a non-singular matrix of order $n$, then $\vert (\text{adj} A)\vert$ is equal to	(IV) $\vert A\vert ^{(n-1)^2}$

Choose the correct answer from the options given below:

Let's match the properties of adjoint matrices using key formulas.

---

Key formulas to remember:

$|\text{adj } A| = |A|^{n-1}$ where $n$ is the order of matrix

$A \cdot (\text{adj } A) = |A| \cdot I$ where $I$ is identity matrix

$\text{adj}(\text{adj } A) = |A|^{n-2} \cdot A$ for non-singular matrices

---

(A) Find $|A(\text{adj } A)|$

We know that $A(\text{adj } A) = |A| \cdot I$

Taking determinant of both sides:

$|A(\text{adj } A)| = ||A| \cdot I|$

$= |A|^n \cdot |I|$

$= |A|^n \cdot 1$

$= |A|^n$

Match: (A) → (III)

---

(B) Find $|\text{adj}(\text{adj } A)|$

Using the formula: $\text{adj}(\text{adj } A) = |A|^{n-2} \cdot A$

Taking determinant of both sides:

$|\text{adj}(\text{adj } A)| = ||A|^{n-2} \cdot A|$

$= (|A|^{n-2})^n \cdot |A|$

$= |A|^{n(n-2)} \cdot |A|$

$= |A|^{n^2-2n+1}$

$= |A|^{(n-1)^2}$

Match: (B) → (IV)

---

(C) Find $\text{adj}(\text{adj } A)$

This is a direct formula:

$\text{adj}(\text{adj } A) = |A|^{n-2} \cdot A$

Match: (C) → (II)

---

(D) Find $|\text{adj } A|$

This is a direct formula:

$|\text{adj } A| = |A|^{n-1}$

Match: (D) → (I)

---

Final Answer: (A) → (III), (B) → (IV), (C) → (II), (D) → (I)