If the matrix is:
\(\left| \begin{array}{ccc} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{array} \right|>0, \text{ then } abc>?\)
\(\left| \begin{array}{ccc} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{array} \right|>0, \text{ then } abc>?\)
Show Hint
- \( 1 \)
- \( -8 \)
- \( 8 \)
- \( 3 \)
The Correct Option is B
Solution and Explanation
To determine the condition for \( abc \) given the inequality:
\[ \left| \begin{array}{ccc} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{array} \right| > 0, \]
we first compute the determinant of the matrix:
\[ \begin{vmatrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{vmatrix}. \]
Expanding the determinant along the first row:
\[ \begin{vmatrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{vmatrix} = a \begin{vmatrix} b & 1 \\ 1 & c \end{vmatrix} - 1 \begin{vmatrix} 1 & 1 \\ 1 & c \end{vmatrix} + 1 \begin{vmatrix} 1 & b \\ 1 & 1 \end{vmatrix}. \]
Calculating the \( 2 \times 2 \) determinants:
\[ \begin{vmatrix} b & 1 \\ 1 & c \end{vmatrix} = bc - 1, \] \[ \begin{vmatrix} 1 & 1 \\ 1 & c \end{vmatrix} = c - 1, \] \[ \begin{vmatrix} 1 & b \\ 1 & 1 \end{vmatrix} = 1 - b. \]
Substituting these back into the expression:
\[ a(bc - 1) - 1(c - 1) + 1(1 - b) = abc - a - c + 1 + 1 - b. \]
Simplifying:
\[ abc - a - b - c + 2. \]
Thus, the inequality becomes:
\[ abc - a - b - c + 2 > 0. \]
Rearranging terms:
\[ abc > a + b + c - 2. \]
To find a lower bound for \( abc \), we consider the case when \( a = b = c \). Let \( a = b = c = k \). Substituting into the inequality:
\[ k^3 > 3k - 2. \]
Rearranging:
\[ k^3 - 3k + 2 > 0. \]
Factoring the cubic equation:
\[ k^3 - 3k + 2 = (k - 1)^2 (k + 2). \]
The roots of the equation \( k^3 - 3k + 2 = 0 \) are \( k = 1 \) (double root) and \( k = -2 \). The inequality \( k^3 - 3k + 2 > 0 \) holds for \( k > -2 \) and \( k \neq 1 \).
For \( k > 1 \), \( k^3 - 3k + 2 > 0 \) is satisfied. For \( k < -2 \), the inequality is not satisfied. Therefore, the minimum value of \( abc \) occurs when \( a = b = c = -2 \):
\[ abc = (-2)^3 = -8. \]
Thus, the inequality \( abc > -8 \) must hold.
Therefore, the correct option is:
\[ \boldsymbol{-8} \]
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