If $\begin{bmatrix}ab&cd\\a+c&b+d\end{bmatrix}=\begin{bmatrix}2&-3\\4&1\end{bmatrix}$ where a, b,c,d are integers, then which of the following are true?

(A) $a + d = 0$
(B) $b + d = 3$
(C) $b+d=1$
(D) $c+d=2$

Choose the correct answer from the options given below:

(A), (C) and (D) only

The correct answer is Option (2) → (A), (C) and (D) only

$\begin{pmatrix}ab & cd\\ a+c & b+d\end{pmatrix} = \begin{pmatrix}2 & -3\\ 4 & 1\end{pmatrix}$

$ab=2,\quad cd=-3,\quad a+c=4,\quad b+d=1$

$ab=2$

$a=1,\; b=2$

$a=2,\; b=1$

$a=-1,\; b=-2$

$a=-2,\; b=-1$

Using $a+c=4$

$c=4-a$

Using $b+d=1$

$d=1-b$

Using $cd=-3$

$(4-a)(1-b)=-3$

Substitute possible $(a,b)$:

$a=2,\; b=1$

$c=4-2=2,\quad d=1-1=0$

$cd=0\neq-3$

$a=1,\; b=2$

$c=4-1=3,\quad d=1-2=-1$

$cd=3(-1)=-3$ ✓

Thus

$a=1,\; b=2,\; c=3,\; d=-1$

$a+d=1+(-1)=0$

$b+d=2+(-1)=1$

$c+d=3+(-1)=2$

Correct options: A, C, D