To solve the expression \((-2) B A + (8) C D\), we need to perform matrix multiplication and scalar multiplication. First, calculate the product \(B A\) and then multiply the result by \(-2\). Next, calculate the product \(C D\) and multiply the result by \(8\). Finally, add the two resulting matrices. Ensure that the dimensions of the matrices are compatible for multiplication.
To solve the problem, we need to compute the expression \((-2)BA + (8)CD\) using the given matrices \(A\), \(B\), \(C\), and \(D\).
First, we need to find the product \(BA\).
Given:
\[
B = \begin{bmatrix} -5 & 1 \\ 3 & 7 \end{bmatrix}, \quad A = \begin{bmatrix} 5 & -2 & 4 \\ 0 & 6 & -5 \end{bmatrix}
\]
The product \(BA\) is calculated as follows:
\[
BA = \begin{bmatrix} -5 & 1 \\ 3 & 7 \end{bmatrix} \begin{bmatrix} 5 & -2 & 4 \\ 0 & 6 & -5 \end{bmatrix} = \begin{bmatrix} (-5)(5) + (1)(0) & (-5)(-2) + (1)(6) & (-5)(4) + (1)(-5) \\ (3)(5) + (7)(0) & (3)(-2) + (7)(6) & (3)(4) + (7)(-5) \end{bmatrix}
\]
\[
= \begin{bmatrix} -25 & 10 + 6 & -20 - 5 \\ 15 & -6 + 42 & 12 - 35 \end{bmatrix} = \begin{bmatrix} -25 & 16 & -25 \\ 15 & 36 & -23 \end{bmatrix}
\]
Now, multiply the result by \(-2\):
\[
(-2)BA = (-2) \begin{bmatrix} -25 & 16 & -25 \\ 15 & 36 & -23 \end{bmatrix} = \begin{bmatrix} 50 & -32 & 50 \\ -30 & -72 & 46 \end{bmatrix}
\]
Next, we need to find the product \(CD\).
Given:
\[
C = \begin{bmatrix} -1 & 0 & 2 \\ 6 & -4 & 1 \\ -2 & 5 & 3 \end{bmatrix}, \quad D = \begin{bmatrix} 4 & -3 \\ 0 & -1 \\ 1 & 3 \end{bmatrix}
\]
The product \(CD\) is calculated as follows:
\[
CD = \begin{bmatrix} -1 & 0 & 2 \\ 6 & -4 & 1 \\ -2 & 5 & 3 \end{bmatrix} \begin{bmatrix} 4 & -3 \\ 0 & -1 \\ 1 & 3 \end{bmatrix} = \begin{bmatrix} (-1)(4) + (0)(0) + (2)(1) & (-1)(-3) + (0)(-1) + (2)(3) \\ (6)(4) + (-4)(0) + (1)(1) & (6)(-3) + (-4)(-1) + (1)(3) \\ (-2)(4) + (5)(0) + (3)(1) & (-2)(-3) + (5)(-1) + (3)(3) \end{bmatrix}
\]
\[
= \begin{bmatrix} -4 + 2 & 3 + 6 \\ 24 + 1 & -18 + 4 + 3 \\ -8 + 3 & 6 - 5 + 9 \end{bmatrix} = \begin{bmatrix} -2 & 9 \\ 25 & -11 \\ -5 & 10 \end{bmatrix}
\]
Now, multiply the result by \(8\):
\[
(8)CD = 8 \begin{bmatrix} -2 & 9 \\ 25 & -11 \\ -5 & 10 \end{bmatrix} = \begin{bmatrix} -16 & 72 \\ 200 & -88 \\ -40 & 80 \end{bmatrix}
\]
Now, add the matrices \((-2)BA\) and \((8)CD\):
\[
(-2)BA + (8)CD = \begin{bmatrix} 50 & -32 & 50 \\ -30 & -72 & 46 \end{bmatrix} + \begin{bmatrix} -16 & 72 \\ 200 & -88 \\ -40 & 80 \end{bmatrix}
\]
Since the matrices have different dimensions, the addition is not possible. Therefore, the expression is not defined.
The answer is not defined. \(\boxed{\text{B}}\)
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