If A is a non-singular square matrix of order n, then which of the following are TRUE?
A. \( |adjA|=|A|^{n-1} \)
B. \( |adjA|=|A|^{n-2} \)
C. \( |adj(adjA)|=|A|^{(n-2)^{2}} \)
D. \( |adj(adjA)|=|A|^{(n-1)^{2}} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → A and D only
Since \( A(\mathrm{adj}\,A)=|A|\,I \), taking determinants of these n × n matrices:
\( |A|\,|\mathrm{adj}\,A|=|A|^{n} \)
\( |\mathrm{adj}\,A|=|A|^{n-1}\quad(|A|\ne 0) \)
The same result for the matrix adj A, whose determinant is \( |A|^{n-1} \), gives
\( |\mathrm{adj}(\mathrm{adj}\,A)|=|\mathrm{adj}\,A|^{n-1} \)
\( =\left(|A|^{n-1}\right)^{n-1}=|A|^{(n-1)^{2}} \)
So \( |\mathrm{adj}\,A|=|A|^{n-1} \) and \( |\mathrm{adj}(\mathrm{adj}\,A)|=|A|^{(n-1)^{2}} \).
Checking the statements
(A) \( |adjA|=|A|^{n-1} \) — Correct. The power is n − 1.
(B) \( |adjA|=|A|^{n-2} \) — Incorrect. The power is n − 1, not n − 2.
(C) \( |adj(adjA)|=|A|^{(n-2)^{2}} \) — Incorrect. The power is \( (n-1)^{2} \), not \( (n-2)^{2} \).
(D) \( |adj(adjA)|=|A|^{(n-1)^{2}} \) — Correct. The power is \( (n-1)^{2} \).
Tip: Do not confuse \( |\mathrm{adj}(\mathrm{adj}\,A)|=|A|^{(n-1)^{2}} \) with the matrix result \( \mathrm{adj}(\mathrm{adj}\,A)=|A|^{n-2}A \); the power n − 2 belongs to the matrix, not the determinant.