For the system \(\begin{bmatrix}2 & -3\\-4 & 6\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}5\\-10\end{bmatrix}\), which of the following statements are correct?
(A) The system has no solution.
(B) The system is consistent.
(C) It has infinitely many solutions.
(D) It has a unique solution.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (B) and (C) only
The system is \(2x - 3y = 5\) and \(-4x + 6y = -10\). The determinant of the coefficient matrix is
\( \displaystyle |A| = \begin{vmatrix}2 & -3\\-4 & 6\end{vmatrix} = 12 - 12 = 0 \)
so there is no unique solution. Next,
\( \displaystyle (\mathrm{adj}\,A)B = \begin{bmatrix}6 & 3\\4 & 2\end{bmatrix}\begin{bmatrix}5\\-10\end{bmatrix} = \begin{bmatrix}0\\0\end{bmatrix} \)
When \(|A| = 0\) and \((\mathrm{adj}\,A)B = O\), the system is consistent with infinitely many solutions. Indeed, the second equation is −2 times the first, so both equations describe the same line.
So the system is consistent and has infinitely many solutions.
Checking the statements
(A) The system has no solution. — Incorrect. Every point of the line \(2x - 3y = 5\), for example (4, 1), is a solution.
(B) The system is consistent. — Correct. It has solutions.
(C) It has infinitely many solutions. — Correct. The two equations represent the same line.
(D) It has a unique solution. — Incorrect. A unique solution needs \(|A| \ne 0\), but here \(|A| = 0\).