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
Equal matrices have equal corresponding entries:
\(ab = 2,\quad cd = -3\)
\(a + c = 4,\quad b + d = 1\)
So \(c = 4 - a\) and \(d = 1 - b\). The integer pairs with \(ab = 2\) are tested against \(cd = -3\):
\(a = 1,\ b = 2:\quad c = 3,\ d = -1,\quad cd = -3\)
\(a = 2,\ b = 1:\quad c = 2,\ d = 0,\quad cd = 0\)
\(a = -1,\ b = -2:\quad c = 5,\ d = 3,\quad cd = 15\)
\(a = -2,\ b = -1:\quad c = 6,\ d = 2,\quad cd = 12\)
Only the first pair works.
So \(a = 1\), \(b = 2\), \(c = 3\), \(d = -1\), and \(b + d = 1\).
Checking the statements
(A) \(a + d = 0\) — Correct. \(1 + (-1) = 0\).
(B) \(b + d = 3\) — Incorrect. \(b + d\) is the bottom-right entry, 1.
(C) \(b + d = 1\) — Correct. \(2 + (-1) = 1\).
(D) \(c + d = 2\) — Correct. \(3 + (-1) = 2\).