If A is an invertible matrix, then which of the following statement(s) is/are TRUE?
(A) $|A^{-1}| = |A|$
(B) $(A^{-1})^{-1} = A$
(C) $A^{-1} =\frac{adj\, A}{|A|}$
(D) $(A^T)^{-1}= (A^{-1})^T$
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (B), (C) and (D) only
Given that A is an invertible matrix:
(A) $|A^{-1}| = |A|$ → ✖ False Because $|A^{-1}| = \frac{1}{|A|}$
(B) $(A^{-1})^{-1} = A$ → ✔ True
(C) $A^{-1} = \frac{\text{adj }A}{|A|}$ → ✔ True (Standard formula for the inverse of a matrix)
(D) $(A^T)^{-1} = (A^{-1})^T$ → ✔ True (Property of transposes and inverses)
Final Answer:
(B), (C), and (D)