Let the matrix $A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$. Then which of the following are true?

(A) adj $A = \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix}$

(B) det $(A) = 5$

(C) det (adjA) = 25

(D) If $A^3 = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then $a + b = c + d$

Choose the correct answer from the options given below:

Given: $A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$

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Checking Statement (A): adj $A = \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix}$

For a 2×2 matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the adjoint is adj $A = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$

adj $A = \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix}$

Statement (A) is TRUE.

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Checking Statement (B): det(A) = 5

For a 2×2 matrix, det $\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$

det(A) $= (2)(4) - (3)(1)$

$= 8 - 3$

$= 5$

Statement (B) is TRUE.

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Checking Statement (C): det(adj A) = 25

For any n×n matrix, det(adj A) = [det(A)]$^{n-1}$

Since the matrix is 2×2:

det(adj A) $= [\text{det}(A)]^{2-1}$

$= [\text{det}(A)]^1$

$= 5$

Statement (C) is FALSE.

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Checking Statement (D): If $A^3 = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then $a + b = c + d$

Calculate $A^2$:

$A^2 = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$

$= \begin{bmatrix} (2)(2) + (3)(1) & (2)(3) + (3)(4) \\ (1)(2) + (4)(1) & (1)(3) + (4)(4) \end{bmatrix}$

$= \begin{bmatrix} 7 & 18 \\ 6 & 19 \end{bmatrix}$

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Calculate $A^3 = A^2 \times A$:

$A^3 = \begin{bmatrix} 7 & 18 \\ 6 & 19 \end{bmatrix} \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$

$= \begin{bmatrix} (7)(2) + (18)(1) & (7)(3) + (18)(4) \\ (6)(2) + (19)(1) & (6)(3) + (19)(4) \end{bmatrix}$

$= \begin{bmatrix} 32 & 93 \\ 31 & 94 \end{bmatrix}$

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$a + b = 32 + 93 = 125$

$c + d = 31 + 94 = 125$

Therefore $a + b = c + d$

Statement (D) is TRUE.

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Statements (A), (B), and (D) are true.