If `[{:(x -y ,4),(z + 6 , x + y ):}] = [{:(8, w),(0,0):}]` , then the value of (x + y + w + z) is

Options

A. 0
B. 2
C. 1
D. `-2`

Answer

To solve the equation given in the question, we need to equate the corresponding elements of the two matrices. The matrices are:

\[ \begin{pmatrix} x - y & 4 \\ z + 6 & x + y \end{pmatrix} = \begin{pmatrix} 8 & w \\ 0 & 0 \end{pmatrix} \]

From this, we can set up the following equations based on the equality of the corresponding elements:

1. \( x - y = 8 \) (from the first element)
2. \( 4 = w \) (from the second element)
3. \( z + 6 = 0 \) (from the third element)
4. \( x + y = 0 \) (from the fourth element)

Now, we will solve these equations step by step:

### Step 1: Solve for \( w \)
From equation 2:
\[ w = 4 \]

### Step 2: Solve for \( z \)
From equation 3:
\[ z + 6 = 0 \implies z = -6 \]

### Step 3: Solve for \( x \) and \( y \)
Now, we have two equations involving \( x \) and \( y \):
1. \( x - y = 8 \) (equation 1)
2. \( x + y = 0 \) (equation 4)

We can add these two equations to eliminate \( y \):
\[ (x - y) + (x + y) = 8 + 0 \]
This simplifies to:
\[ 2x = 8 \implies x = 4 \]

Now, substitute \( x = 4 \) into equation 4 to find \( y \):
\[ 4 + y = 0 \implies y = -4 \]

### Step 4: Calculate \( x + y + w + z \)
Now we can find the value of \( x + y + w + z \):
\[ x + y + w + z = 4 + (-4) + 4 + (-6) \]
Calculating this gives:
\[ = 0 + 4 - 6 = -2 \]

Thus, the final answer is:
\[ \boxed{-2} \]