Let A be a square matrix of order n, then which of the following are TRUE?

(A) $|adj\, A| = |A|^{n-1}$
(B) $|A.\, adj\, A| = |A|^n$
(C) $A (adj\, A)= |A|$
(D) $|KA| = K|A|$
(E) $|A^{-1}=\frac{1}{|A|},|A| ≠ 0$

Choose the correct answer from the options given below:

(A), (B) and (E) only

The correct answer is Option (1) → (A), (B) and (E) only

Given a square matrix $A$ of order $n$:

(A) $\;|\text{adj}A| = |A|^{n-1}$ — True (since determinant of adjugate is $|A|^{n-1}$).

(B) $\;|A\cdot \text{adj}A| = |A|^n$ — True, because $|A\cdot \text{adj}A| = |A|\;|\text{adj}A| = |A|\cdot|A|^{n-1}=|A|^n$.

(C) $\;A(\text{adj}A)=|A|$ — False, since $A(\text{adj}A)=|A|I$ (a matrix), not a scalar.

(D) $\;|KA|=K|A|$ — False, the correct formula is $|KA|=K^n|A|$.

(E) $\;|A^{-1}|=\frac{1}{|A|},\;|A|\neq0$ — True.

Correct statements: (A), (B), and (E)