If A,B,C are the angles of triangle ABC, then the minimum value of `|{:(-2,cos C , cos B),(cos C , -1, cos A ) , (cos B , cos A , -1):}|` is equal to :

Options

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

Answer

To find the minimum value of the determinant

\[ D = \begin{vmatrix} -2 & \cos C & \cos B \\ \cos C & -1 & \cos A \\ \cos B & \cos A & -1 \end{vmatrix} \]

we will follow these steps:

### Step 1: Expand the Determinant

We will use the formula for the determinant of a 3x3 matrix. The determinant can be expanded as follows:

\[ D = -2 \begin{vmatrix} -1 & \cos A \\ \cos A & -1 \end{vmatrix} - \cos C \begin{vmatrix} \cos C & \cos A \\ \cos B & -1 \end{vmatrix} + \cos B \begin{vmatrix} \cos C & -1 \\ \cos B & \cos A \end{vmatrix} \]

### Step 2: Calculate the 2x2 Determinants

1. For the first determinant:

\[ \begin{vmatrix} -1 & \cos A \\ \cos A & -1 \end{vmatrix} = (-1)(-1) - (\cos A)(\cos A) = 1 - \cos^2 A = \sin^2 A \]

2. For the second determinant:

\[ \begin{vmatrix} \cos C & \cos A \\ \cos B & -1 \end{vmatrix} = (\cos C)(-1) - (\cos A)(\cos B) = -\cos C - \cos A \cos B \]

3. For the third determinant:

\[ \begin{vmatrix} \cos C & -1 \\ \cos B & \cos A \end{vmatrix} = (\cos C)(\cos A) - (-1)(\cos B) = \cos C \cos A + \cos B \]

### Step 3: Substitute Back into the Determinant

Now substituting these back into the determinant expression:

\[ D = -2 \sin^2 A + \cos C (\cos C + \cos A \cos B) + \cos B (\cos C \cos A + \cos B) \]

### Step 4: Simplify the Expression

After simplifying, we get:

\[ D = -2 \sin^2 A + \cos^2 C + \cos C \cos A \cos B + \cos B \cos C \cos A + \cos^2 B \]

### Step 5: Use Trigonometric Identities

Using the identity \(\sin^2 A + \cos^2 A = 1\), we can express \(\sin^2 A\) in terms of \(\cos^2 A\):

\[ D = -2(1 - \cos^2 A) + \cos^2 C + 2 \cos A \cos B + \cos^2 B \]

### Step 6: Find the Minimum Value

To find the minimum value, we can analyze the expression. The minimum value occurs when \(\sin^2 A\), \(\cos^2 B\), and \(\cos^2 C\) take their minimum values.

Since \(\sin^2 A\) can be 0 and \(\cos^2 A\), \(\cos^2 B\), and \(\cos^2 C\) can also be 0, we can substitute these values into the expression:

\[ D_{min} = -2(0) + 0 + 0 + 0 = -2 \]

Thus, the minimum value of the determinant is:

\[ \boxed{-2} \]