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) \(A^{-1} =\displaystyle\frac{\mathrm{adj}\, A}{|A|}\)
(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
Since A is invertible, \(|A| \ne 0\) and
\(AA^{-1} = A^{-1}A = I\)
Taking determinants:
\(|A|\,|A^{-1}| = 1\)
\( \displaystyle |A^{-1}| = \frac{1}{|A|} \)
Taking transposes, with the reversal law \((PQ)^T = Q^TP^T\):
\((A^{-1})^T A^T = I^T = I\)
\((A^T)^{-1} = (A^{-1})^T\)
So \(|A^{-1}| = \frac{1}{|A|}\), the inverse of \(A^{-1}\) is A, \(A^{-1} = \frac{1}{|A|}\,\mathrm{adj}\,A\), and \((A^T)^{-1} = (A^{-1})^T\).
Checking the statements
(A) \(|A^{-1}| = |A|\) — Incorrect. \(|A^{-1}| = \frac{1}{|A|}\), equal to \(|A|\) only when \(|A| = \pm 1\).
(B) \((A^{-1})^{-1} = A\) — Correct. \(A^{-1}A = I\), so A is the inverse of \(A^{-1}\).
(C) \(A^{-1} =\displaystyle\frac{\mathrm{adj}\, A}{|A|}\) — Correct. This is the adjoint formula for the inverse, valid as \(|A| \ne 0\).
(D) \((A^T)^{-1}= (A^{-1})^T\) — Correct. \((A^{-1})^T A^T = (AA^{-1})^T = I\).