If \( A=\begin{bmatrix}1 & -3\\2 & 0\end{bmatrix} \), then which of the following statement(s) is/are TRUE?
(A) Matrix A is non-singular
(B) \( |3A|=54 \)
(C) \( |adjA|=36 \)
(D) \( A^2=\begin{bmatrix}-5 & -3\\2 & -6\end{bmatrix} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (D) only
\( |A|=(1)(0)-(-3)(2)=6 \)
For a matrix of order n, \( |kA|=k^n|A| \) and \( |\mathrm{adj}\,A|=|A|^{n-1} \). Here n = 2:
\( |3A|=3^2\times 6=54 \)
\( |\mathrm{adj}\,A|=6^{1}=6 \)
\( \displaystyle A^2=\begin{bmatrix}1 & -3\\2 & 0\end{bmatrix}\begin{bmatrix}1 & -3\\2 & 0\end{bmatrix} \)
\( \displaystyle =\begin{bmatrix}1-6 & -3+0\\2+0 & -6+0\end{bmatrix} \)
\( \displaystyle =\begin{bmatrix}-5 & -3\\2 & -6\end{bmatrix} \)
So \( |A|=6 \), \( |3A|=54 \), \( |\mathrm{adj}\,A|=6 \) and \( A^2=\begin{bmatrix}-5 & -3\\2 & -6\end{bmatrix} \).
Checking the statements
(A) Matrix A is non-singular — Correct. \( |A|=6\ne 0 \).
(B) \( |3A|=54 \) — Correct. \( |3A|=3^2\times 6=54 \).
(C) \( |adjA|=36 \) — Incorrect. For a 2 × 2 matrix \( |\mathrm{adj}\,A|=|A|=6 \), not 36.
(D) \( A^2=\begin{bmatrix}-5 & -3\\2 & -6\end{bmatrix} \) — Correct. Row by column: 1 − 6 = −5, −3 + 0 = −3, 2 + 0 = 2 and −6 + 0 = −6.