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:

The system can be written as:

$\begin{bmatrix} 2 & -3 \\ -4 & 6 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ -10 \end{bmatrix}$

This gives two equations:

$2x - 3y = 5$ ... (1)

$-4x + 6y = -10$ ... (2)

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Multiply equation (1) by $-2$:

$-2(2x - 3y) = -2(5)$

$-4x + 6y = -10$

This is identical to equation (2). Both equations represent the same line.

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The determinant of the coefficient matrix:

$\text{Det} = (2)(6) - (-3)(-4)$

$= 12 - 12$

$= 0$

When the determinant is zero, the system either has no solution or infinitely many solutions.

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Since both equations are identical (not contradictory), every point on the line $2x - 3y = 5$ is a solution.

Therefore:

(A) The system has no solution - False

(B) The system is consistent - True (has at least one solution)

(C) It has infinitely many solutions - True

(D) It has a unique solution - False

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The correct statements are (B) and (C) only.