If $A =\begin{bmatrix}1&2\\4&5\end{bmatrix}$, then

Match List-I with List-II

Choose the correct answer from the options given below:

(A)-(III), (B)-(I), (C)-(II), (D)-(IV)

The correct answer is Option (2) → (A)-(III), (B)-(I), (C)-(II), (D)-(IV)

Given:

$A = \begin{bmatrix} 1 & 2 \\ 4 & 5 \end{bmatrix}$

$\det(A) = (1)(5) - (2)(4) = 5 - 8 = -3$

⇒ (A) $\det(A) = -3$ → (III)

$\det(A^{-1}) = \frac{1}{\det(A)} = \frac{1}{-3} = -\frac{1}{3}$

⇒ (B) $\det(A^{-1}) = -\frac{1}{3}$ → (I)

$\det(2A) = 2^2 \det(A) = 4(-3) = -12$

⇒ (C) $\det(2A) = -12$ → (II)

$\det(3A^T) = 3^2 \det(A^T) = 9\det(A) = 9(-3) = -27$

⇒ (D) $\det(3A^T) = -27$ → (IV)

Final Matching:

(A) → (III), (B) → (I), (C) → (II), (D) → (IV)