设A为n阶矩阵,满足A乘以A的转置矩阵=E,|A| |A| = -1
问题描述：
设A为n阶矩阵,满足A乘以A的转置矩阵=E,|A| |A| = -1
∵|A| ≠ 0∴A存在逆矩阵,∵A * A^T = 1,∴A⁻¹ = A^T
|A + E| = |A + AA⁻¹| = |A(E + A⁻¹)| = |A| |E + A^T| = - |E^T + A^T| = - |(E + A)^T| = - |E + A|
=> 2|A + E| = 0
=> |A + E| = 0
答
|A^T| = |A|这是行列式的性质
转置行列式值不变