If `B = [{:(-2, 2, 0),(3,1,4):}]; C= [{:(2,0,-2),(7,1,6):}]and 2 A - 3 B + 5 C = 0,` then the matrix A is :

Options

A. `[{:(-8, 3, 5),(-13, -1, -9):}]`
B. `[{:(16,-6, - 10),( 26, 2, 18):}]`
C. `[{:(8, -3, -5),(13,1,9):}]`
D. `[{:(7, 4, -1),( 2, -3, 5):}]`

Answer

To solve the equation \(2A - 3B + 5C = 0\) for the matrix \(A\), we will follow these steps:

### Step 1: Write down the matrices \(B\) and \(C\)

Given:
\[ B = \begin{pmatrix} -2 & 2 & 0 \\ 3 & 1 & 4 \end{pmatrix}, \quad C = \begin{pmatrix} 2 & 0 & -2 \\ 7 & 1 & 6 \end{pmatrix} \]

### Step 2: Calculate \(-3B\)

To find \(-3B\), we multiply each element of \(B\) by \(-3\):
\[ -3B = -3 \begin{pmatrix} -2 & 2 & 0 \\ 3 & 1 & 4 \end{pmatrix} = \begin{pmatrix} 6 & -6 & 0 \\ -9 & -3 & -12 \end{pmatrix} \]

### Step 3: Calculate \(5C\)

Next, we calculate \(5C\) by multiplying each element of \(C\) by \(5\):
\[ 5C = 5 \begin{pmatrix} 2 & 0 & -2 \\ 7 & 1 & 6 \end{pmatrix} = \begin{pmatrix} 10 & 0 & -10 \\ 35 & 5 & 30 \end{pmatrix} \]

### Step 4: Substitute \(-3B\) and \(5C\) into the equation

Now we substitute \(-3B\) and \(5C\) into the original equation:
\[ 2A - 3B + 5C = 0 \implies 2A + \begin{pmatrix} 6 & -6 & 0 \\ -9 & -3 & -12 \end{pmatrix} + \begin{pmatrix} 10 & 0 & -10 \\ 35 & 5 & 30 \end{pmatrix} = 0 \]

### Step 5: Combine the matrices

Next, we combine the matrices:
\[ 2A + \begin{pmatrix} 6 + 10 & -6 + 0 & 0 - 10 \\ -9 + 35 & -3 + 5 & -12 + 30 \end{pmatrix} = 0 \]
This simplifies to:
\[ 2A + \begin{pmatrix} 16 & -6 & -10 \\ 26 & 2 & 18 \end{pmatrix} = 0 \]

### Step 6: Isolate \(2A\)

Now, we isolate \(2A\):
\[ 2A = -\begin{pmatrix} 16 & -6 & -10 \\ 26 & 2 & 18 \end{pmatrix} = \begin{pmatrix} -16 & 6 & 10 \\ -26 & -2 & -18 \end{pmatrix} \]

### Step 7: Solve for \(A\)

Finally, we divide each element by \(2\) to find \(A\):
\[ A = \frac{1}{2} \begin{pmatrix} -16 & 6 & 10 \\ -26 & -2 & -18 \end{pmatrix} = \begin{pmatrix} -8 & 3 & 5 \\ -13 & -1 & -9 \end{pmatrix} \]

### Final Answer

Thus, the matrix \(A\) is:
\[ A = \begin{pmatrix} -8 & 3 & 5 \\ -13 & -1 & -9 \end{pmatrix} \]