If `|{:(4,-4,0),(a,b+4,c),(a,b,c+4):}|=0`, then `a+b+c` is equal to

Options

A. 41
B. 116
C. 628
D. -4

Answer

To solve the problem, we need to evaluate the determinant given by:

\[ \begin{vmatrix} 4 & -4 & 0 \\ a & b+4 & c \\ a & b & c+4 \end{vmatrix} = 0 \]

### Step 1: Expand the Determinant

We will expand the determinant along the first row (R1):

\[ D = 4 \begin{vmatrix} b+4 & c \\ b & c+4 \end{vmatrix} - (-4) \begin{vmatrix} a & c \\ a & c+4 \end{vmatrix} + 0 \]

### Step 2: Calculate the 2x2 Determinants

Now we calculate the two 2x2 determinants:

1. For the first determinant:
\[ \begin{vmatrix} b+4 & c \\ b & c+4 \end{vmatrix} = (b+4)(c+4) - bc = bc + 4b + 4c + 16 - bc = 4b + 4c + 16 \]

2. For the second determinant:
\[ \begin{vmatrix} a & c \\ a & c+4 \end{vmatrix} = a(c+4) - ac = ac + 4a - ac = 4a \]

### Step 3: Substitute Back into the Determinant

Now substituting these back into the expression for D:

\[ D = 4(4b + 4c + 16) + 4(4a) \]

### Step 4: Simplify the Expression

Now simplify the expression:

\[ D = 16b + 16c + 64 + 16a \]

### Step 5: Set the Determinant to Zero

Since we are given that the determinant equals zero:

\[ 16a + 16b + 16c + 64 = 0 \]

### Step 6: Factor Out Common Terms

We can factor out 16:

\[ 16(a + b + c + 4) = 0 \]

### Step 7: Solve for \(a + b + c\)

Since \(16 \neq 0\), we can divide both sides by 16:

\[ a + b + c + 4 = 0 \]

Thus, we have:

\[ a + b + c = -4 \]

### Final Answer

So, the value of \(a + b + c\) is:

\[ \boxed{-4} \]