If \( A \) is a square matrix of order \( n\times n \), then which of the following are correct ?
(A) \( |k\ A|=k^n|A|,\ k \) is a constant
(B) \( |k\ adjA|=k^n|adjA|^{n-1},\ k \) is a constant
(C) \( |adj\ adjA|=|A|^{(n-1)^2} \)
(D) \( A\ (adjA)=|A|I \) , \( I \) is an identity matrix of order \( n \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (C) and (D) only
For a square matrix of order n, multiplying the matrix by k multiplies each of its n rows by k, and the adjoint satisfies
\( A(\mathrm{adj}\,A)=|A|I \)
\( |\mathrm{adj}\,A|=|A|^{n-1} \)
So \( |kA|=k^n|A| \), \( |k\,\mathrm{adj}\,A|=k^n|\mathrm{adj}\,A| \), \( |\mathrm{adj}(\mathrm{adj}\,A)|=|A|^{(n-1)^2} \) and \( A(\mathrm{adj}\,A)=|A|I \).
Checking the statements
(A) \( |k\ A|=k^n|A|,\ k \) is a constant — Correct. Each of the n rows is multiplied by k.
(B) \( |k\ adjA|=k^n|adjA|^{n-1},\ k \) is a constant — Incorrect. The value is \( k^n|\mathrm{adj}\,A| \), with no power n − 1 on |adj A|.
(C) \( |adj\ adjA|=|A|^{(n-1)^2} \) — Correct. Applying \( |\mathrm{adj}\,B|=|B|^{n-1} \) twice gives \( (|A|^{n-1})^{n-1} \).
(D) \( A\ (adjA)=|A|I \) , \( I \) is an identity matrix of order \( n \) — Correct. This is the basic property of the adjoint.