题目
Problem
=M(3x4x+172−x)
Find the range of values of x for which the determinant of the matrix M is positive.
(5)
题目中文翻译
=M(3x4x+172−x)
求使矩阵 M 的行列式为正值的 x 的取值范围。
(5)
解答
解法一
思路
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先按二阶矩阵行列式公式展开并化简，再把二次不等式因式分解。找出两个临界值后，根据二次项系数为负判断表达式在两根之间为正；由于题目要求严格大于零，端点不包括在内。
答题过程
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The determinant of M is
===detM(3x)(2−x)−7(4x+1)6x−3x2−28x−7−3x2−22x−7.
For the determinant to be positive,
−3x2−22x−7>0.
Factorising the quadratic gives
===−3x2−22x−7−3x2−21x−x−7−3x(x+7)−(x+7)(−3x−1)(x+7).
Thus the critical values are
x=x=−7,−31.
Since the coefficient of x2 is negative, the quadratic is positive between its roots. In this interval, −3x−1>0 and x+7>0, so their product is positive. Outside this interval the two factors have opposite signs. The endpoints give a zero determinant and are excluded. Therefore,
−7<x<−31.
