If A is a 3-rowed square matrix and \( |\mathrm{A}|=5 \), then \( |\mathrm{adj}\mathrm{A}| \) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 25
For a square matrix of order n, \( |\mathrm{adj}\,\mathrm{A}|=|\mathrm{A}|^{n-1} \). This follows from \( \mathrm{A}(\mathrm{adj}\,\mathrm{A})=|\mathrm{A}|\,\mathrm{I} \): taking determinants,
\( |\mathrm{A}|\,|\mathrm{adj}\,\mathrm{A}|=|\mathrm{A}|^n \)
\( |\mathrm{adj}\,\mathrm{A}|=|\mathrm{A}|^{n-1} \)
Here n = 3:
\( |\mathrm{adj}\,\mathrm{A}|=5^{3-1}=25 \)
Why the other options are incorrect
Option (1) 5 — Incorrect. This is |A| itself, which equals \( |\mathrm{adj}\,\mathrm{A}| \) only for a 2 × 2 matrix.
Option (3) 125 — Incorrect. This is \( |\mathrm{A}|^3=|\mathrm{A}|^n \); the power is n − 1, not n.
Option (4) 225 — Incorrect. 225 is not a power of 5, so it cannot be \( |\mathrm{A}|^{n-1} \).
Tip: Remember the three results for a matrix of order n: \( |k\mathrm{A}|=k^n|\mathrm{A}| \), \( |\mathrm{adj}\,\mathrm{A}|=|\mathrm{A}|^{n-1} \) and \( |\mathrm{adj}(\mathrm{adj}\,\mathrm{A})|=|\mathrm{A}|^{(n-1)^2} \).
NCERT Exemplar:
Determinants
