If `A=[(x,y),(z,w)],B=[(x,-y),(-z,w)]`
and `C=[(-2x,0),(0,-2w)]` then `A+B+C` is a

Options

A. identify matrix
B. null matrix
C. row matrix
D. column matrix

Answer

To solve the problem, we need to perform the matrix addition of \( A \), \( B \), and \( C \). Let's break it down step by step.

### Step 1: Define the matrices
We have the following matrices:
- \( A = \begin{pmatrix} x & y \\ z & w \end{pmatrix} \)
- \( B = \begin{pmatrix} x & -y \\ -z & w \end{pmatrix} \)
- \( C = \begin{pmatrix} -2x & 0 \\ 0 & -2w \end{pmatrix} \)

### Step 2: Add matrices \( A \) and \( B \)
To add matrices \( A \) and \( B \), we add their corresponding elements:
\[ A + B = \begin{pmatrix} x & y \\ z & w \end{pmatrix} + \begin{pmatrix} x & -y \\ -z & w \end{pmatrix} = \begin{pmatrix} x + x & y + (-y) \\ z + (-z) & w + w \end{pmatrix} \]
Calculating the elements:
\[ A + B = \begin{pmatrix} 2x & 0 \\ 0 & 2w \end{pmatrix} \]

### Step 3: Add the result to matrix \( C \)
Now we add the result of \( A + B \) to matrix \( C \):
\[ (A + B) + C = \begin{pmatrix} 2x & 0 \\ 0 & 2w \end{pmatrix} + \begin{pmatrix} -2x & 0 \\ 0 & -2w \end{pmatrix} \]
Adding the corresponding elements:
\[ (A + B) + C = \begin{pmatrix} 2x + (-2x) & 0 + 0 \\ 0 + 0 & 2w + (-2w) \end{pmatrix} \]
Calculating the elements:
\[ (A + B) + C = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

### Step 4: Conclusion
The result of \( A + B + C \) is the null matrix (zero matrix):
\[ A + B + C = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]