What is `|{:(-a^(2),ab,ac),(ab,-b^(2),bc),(ac,bc,-c^(2)):}|` equal to ?

Options

A. `4a^(2) b^(2)`
B. `4b^(2) c^(2)`
C. `4c^(2) a^(2)`
D. `4a^(2) b^(2) c^(2)`

Explanation

To find the value of the determinant

\[ D = \begin{vmatrix} -a^2 & ab & ac \\ ab & -b^2 & bc \\ ac & bc & -c^2 \end{vmatrix} \]

we will follow these steps:

### Step 1: Factor out common terms from the columns
We can factor out \(-a\) from the first column, \(b\) from the second column, and \(c\) from the third column.

\[ D = (-a)(b)(c) \begin{vmatrix} a & b & c \\ b & -b & \frac{bc}{b} \\ c & \frac{bc}{c} & -c \end{vmatrix} \]

This simplifies to:

\[ D = -abc \begin{vmatrix} a & b & c \\ b & -b & c \\ c & b & -c \end{vmatrix} \]

### Step 2: Rewrite the determinant
Now we can rewrite the determinant:

\[ D = -abc \begin{vmatrix} a & b & c \\ b & -b & c \\ c & b & -c \end{vmatrix} \]

### Step 3: Expand the determinant
We will expand the determinant along the first row:

\[ D = -abc \left( a \begin{vmatrix} -b & c \\ b & -c \end{vmatrix} - b \begin{vmatrix} b & c \\ c & -c \end{vmatrix} + c \begin{vmatrix} b & -b \\ c & b \end{vmatrix} \right) \]

### Step 4: Calculate the 2x2 determinants
Calculating the 2x2 determinants:

1. \(\begin{vmatrix} -b & c \\ b & -c \end{vmatrix} = (-b)(-c) - (c)(b) = bc - bc = 0\)

2. \(\begin{vmatrix} b & c \\ c & -c \end{vmatrix} = (b)(-c) - (c)(c) = -bc - c^2 = -bc - c^2\)

3. \(\begin{vmatrix} b & -b \\ c & b \end{vmatrix} = (b)(b) - (-b)(c) = b^2 + bc\)

### Step 5: Substitute back into the determinant
Substituting these values back into the determinant:

\[ D = -abc \left( a(0) - b(-bc - c^2) + c(b^2 + bc) \right) \]

This simplifies to:

\[ D = -abc \left( b(bc + c^2) + c(b^2 + bc) \right) \]

### Step 6: Simplify the expression
Now we can simplify the expression:

\[ D = -abc \left( b^2c + bc^2 + cb^2 + c^2b \right) = -abc(2b^2c + 2bc^2) \]

### Step 7: Final expression
Thus, we have:

\[ D = -2abc(b^2 + c^2) \]

### Final Result
The value of the determinant is:

\[ D = -abc(a^2 + b^2 + c^2) \]