If A and B are square matrices of the same order, then which of the following statements are correct?
(A) \(|A^{-1}| = |A|^{-1}\)
(B) \(\mathrm{adj}\,(A) = |A|A^{-1}\)
(C) \((A + B)^{-1} = B^{-1} + A^{-1}\)
(D) \((AB)^{-1} = B^{-1}A^{-1}\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (D) only
The statements involve inverses, so A and B are taken to be non-singular (their inverses exist).
(A) \(|A^{-1}| = |A|^{-1}\) — Correct. Taking determinants in \(AA^{-1} = I\):
\(|A|\,|A^{-1}| = |I| = 1\)
\( \displaystyle |A^{-1}| = \frac{1}{|A|} = |A|^{-1} \)
(B) \(\mathrm{adj}\,(A) = |A|A^{-1}\) — Correct. By the formula for the inverse,
\( \displaystyle A^{-1} = \frac{1}{|A|}\,\mathrm{adj}\,(A) \)
and multiplying both sides by \(|A|\) gives \(\mathrm{adj}\,(A) = |A|A^{-1}\).
(C) \((A + B)^{-1} = B^{-1} + A^{-1}\) — Incorrect. Take \(A = B = I\):
\( \displaystyle (A + B)^{-1} = (2I)^{-1} = \frac{1}{2}I \)
\(B^{-1} + A^{-1} = I + I = 2I\)
The two are not equal.
(D) \((AB)^{-1} = B^{-1}A^{-1}\) — Correct. This is the reversal law:
\((AB)(B^{-1}A^{-1}) = A(BB^{-1})A^{-1}\)
\(= AA^{-1} = I\)