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:
Answer & explanation
Correct answer: option 1
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)