If \( A=\begin{bmatrix}3 & 0 & -1\\ 2 & 3 & 0\\ 0 & 4 & 1\end{bmatrix} \), then which of the following is/are correct?
A. \( |A|=1 \)
B. \( A \) is singular matrix.
C. \( |adjA|=1 \)
D. \( |adj.(adjA)|=1 \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A, C and D only
Expanding \( |A| \) along the first row,
\( \displaystyle |A|=3(3\cdot1-0\cdot4)-0+(-1)(2\cdot4-3\cdot0) \)
\( \displaystyle =9-8 \)
\( \displaystyle =1 \)
Since \( |A|\ne0 \), A is non-singular. For a matrix of order n = 3,
\( \displaystyle |adj\,A|=|A|^{n-1} \)
\( \displaystyle |adj(adj\,A)|=|A|^{(n-1)^{2}} \)
So
\( \displaystyle |adj\,A|=1^{2}=1 \)
\( \displaystyle |adj(adj\,A)|=1^{4}=1 \)
So \( |A|=1 \), A is non-singular, \( |adj\,A|=1 \) and \( |adj(adj\,A)|=1 \).
Checking the statements
(A) \( |A|=1 \) — Correct. \( |A|=9-8=1 \).
(B) A is singular matrix — Incorrect. \( |A|=1\ne0 \), so A is non-singular.
(C) \( |adjA|=1 \) — Correct. \( |adj\,A|=|A|^{2}=1 \).
(D) \( |adj.(adjA)|=1 \) — Correct. \( |adj(adj\,A)|=|A|^{4}=1 \).