`|(0,b,-c),(-b,0,a),(c,-a,0)|=`

Options

A. 0
B. abc
C. `-abc`
D. none

Answer

To find the value of the determinant

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

we will use the formula for the determinant of a 3x3 matrix. The determinant of a matrix

\[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} \]

is calculated as:

\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our specific matrix, we can identify:

- \( a = 0 \)
- \( b = b \)
- \( c = -c \)
- \( d = -b \)
- \( e = 0 \)
- \( f = a \)
- \( g = c \)
- \( h = -a \)
- \( i = 0 \)

Now, substituting these values into the determinant formula:

\[ D = 0(0 \cdot 0 - a \cdot (-a)) - b(-b \cdot 0 - a \cdot c) + (-c)(-b \cdot (-a) - 0 \cdot c) \]

Calculating each term:

1. The first term is \( 0 \cdot (0 + a^2) = 0 \).
2. The second term is \( -b(0 - ac) = b(ac) \).
3. The third term is \( -c(b \cdot a - 0) = -cab \).

Putting it all together:

\[ D = 0 + bac - cab = bac - cab \]

Since \( bac \) and \( cab \) are the same, we can simplify:

\[ D = 0 \]

Thus, the value of the determinant is:

\[ \boxed{0} \]