If `a ne b ne c` all are positive, then the value of determinant `|(a,b,c),(b,c,a),(c,a,b)|` is

Options

A. Non-negative
B. Non-positive
C. Negative
D. Positive

Answer

To find the value of the determinant

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

we will calculate it step by step.

### Step 1: Write the Determinant
We start by writing the determinant explicitly:

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

### Step 2: Apply the Determinant Formula
Using the formula for a 3x3 determinant, we have:

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

### Step 3: Calculate Each 2x2 Determinant
Now we will calculate each of the 2x2 determinants:

1. For the first term:
\[ \begin{vmatrix} c & a \\ a & b \end{vmatrix} = cb - a^2 \]

2. For the second term:
\[ \begin{vmatrix} b & a \\ c & b \end{vmatrix} = bb - ac = b^2 - ac \]

3. For the third term:
\[ \begin{vmatrix} b & c \\ c & a \end{vmatrix} = ba - c^2 \]

### Step 4: Substitute Back into the Determinant
Substituting these back into the expression for \(D\):

\[ D = a(cb - a^2) - b(b^2 - ac) + c(ba - c^2) \]

### Step 5: Expand the Expression
Now we expand this expression:

\[ D = acb - a^3 - b^3 + abc + abc - c^3 \]

Combining like terms:

\[ D = 3abc - (a^3 + b^3 + c^3) \]

### Step 6: Factor the Expression
We can factor the expression using the identity for the sum of cubes:

\[ D = -(a^3 + b^3 + c^3 - 3abc) = -(a+b+c)(a^2 + b^2 + c^2 - ab - ac - bc) \]

### Step 7: Analyze the Result
Since \(a\), \(b\), and \(c\) are all positive, the term \((a+b+c)\) is positive. The term \((a^2 + b^2 + c^2 - ab - ac - bc)\) is also non-negative (it is the result of the squares of the differences). Therefore, \(D\) is negative.

### Conclusion
Thus, the value of the determinant is:

\[ D < 0 \]