Question

线性代数矩阵，第三题，后面两个怎么证明

Answer 1

(1) n = 2 时，|A1| = - |A| ， |A2| = - |A^T| = - |A| ;
n = 3 时，|A1| = - |A| ， |A2| = - |A^T| = - |A| ;
n = 4 时，|A1| = (-1)^2 |A| = |A|， |A2| = (-1)^2 |A^T| = |A| ;
n = 5 时，|A1| = (-1)^2 |A| = |A|， |A2| = (-1)^2 |A^T| = |A| ;
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|A1| = |A2| = (-1)^[(1/2)n(n-1)] |A| .
(2) A3 相当于 A 变成 A2，再按 A2 方法变化，故
|A3| = (-1)^[(1/2)n(n-1)] (-1)^[(1/2)n(n-1)] |A| = |A|