If `A=[{:(-2," "3),(" "4," "5),(" "1," "-6):}]" and "B=[{:(" "5," "2),(-7," "3),(" "6," "4):}]`, find a matrix C such that `A+B-C=O.`

Options

A. `[{:(3," "5),(3," "8),(7,-2):}]`
B. `[{:(-3," "5),(-3," "8),(7,-2):}]`
C. `[{:(3," "5),(-3," "8),(7,-2):}]`
D. `[{:(3," "5),(-3," "8),(-7,-2):}]`

Explanation

To find the matrix \( C \) such that \( A + B - C = O \), where \( O \) is the null matrix, we can follow these steps:

### Step 1: Define the matrices \( A \) and \( B \)

Given:
\[ A = \begin{pmatrix} -2 & 3 \\ 4 & 5 \\ 1 & -6 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 2 \\ -7 & 3 \\ 6 & 4 \end{pmatrix} \]

### Step 2: Find \( A + B \)

To find \( A + B \), we add the corresponding elements of matrices \( A \) and \( B \):

\[ A + B = \begin{pmatrix} -2 + 5 & 3 + 2 \\ 4 - 7 & 5 + 3 \\ 1 + 6 & -6 + 4 \end{pmatrix} \]

Calculating each element:
- First row: \( -2 + 5 = 3 \) and \( 3 + 2 = 5 \)
- Second row: \( 4 - 7 = -3 \) and \( 5 + 3 = 8 \)
- Third row: \( 1 + 6 = 7 \) and \( -6 + 4 = -2 \)

Thus,
\[ A + B = \begin{pmatrix} 3 & 5 \\ -3 & 8 \\ 7 & -2 \end{pmatrix} \]

### Step 3: Set up the equation \( A + B - C = O \)

Since \( O \) is the null matrix, we have:
\[ A + B - C = O \implies C = A + B \]

### Step 4: Write the expression for \( C \)

From the previous step, we have:
\[ C = \begin{pmatrix} 3 & 5 \\ -3 & 8 \\ 7 & -2 \end{pmatrix} \]

### Conclusion

Thus, the matrix \( C \) is:
\[ C = \begin{pmatrix} 3 & 5 \\ -3 & 8 \\ 7 & -2 \end{pmatrix} \]