If A is an invertible matrix of order \(3 \times 3\), then the value of \(|\mathrm{adj}(A)|\) is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(|A|^2\)
For any square matrix of order n:
\(A(\mathrm{adj}\,A) = |A|\,I\)
Taking determinants, with \(|kI| = k^3\) for a 3 × 3 matrix:
\(|A|\,|\mathrm{adj}\,A| = |A|^3\)
A is invertible, so \(|A| \ne 0\) and we may divide:
\(|\mathrm{adj}\,A| = |A|^2\)
This is the result \(|\mathrm{adj}\,A| = |A|^{n-1}\) with n = 3.
Why the other options are incorrect
Option (1) \(|A|\) — Incorrect. \(|A|^{n-1} = |A|\) only for n = 2.
Option (3) \(|A|^3\) — Incorrect. This is \(|\,|A|\,I\,|\), the determinant of \(A(\mathrm{adj}\,A)\), before dividing by \(|A|\).
Option (4) \(\left|A^{-1}\right|\) — Incorrect. \(\left|A^{-1}\right| = \frac{1}{|A|}\), which is not \(|A|^2\) in general.
NCERT (2026-27 edition):
Adjoint and Inverse of a Matrix
