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:
Answer & explanation
Correct answer: option 2
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