设A为可逆矩阵,且每行元素之和都有等于常数a≠0,证明A-1 (-1为)A右上角的 的每一行元素之和都等于a-1 设A是n阶可逆矩阵 若A的每一行元素之和为c 求证A^-1每...
\u8bben\u9636\u53ef\u9006\u77e9\u9635A\u4e2d\u6bcf\u884c\u4e4b\u548c\u5143\u7d20\u4e3a\u5e38\u6570a\uff0c\u8bc1\u660eA^(-1)\u7684\u6bcf\u884c\u5143\u7d20\u4e4b\u548c\u4e3aa^(-1)\u8bc1\u660e\uff1a
\u4ee4\u5217\u5411\u91cfx=\uff081 1.......1)^-1
\u5219\u7531\u9898\u610f\u53ef\u77e5Ax=\uff08a a...........a)^-1
\u4e0a\u5f0f\u4e24\u8fb9\u540c\u4e58A^-1\u53ef\u5f97
x=A^(-1)*(a a\u2026\u2026a)^-1\uff0c\u4e24\u8fb9\u540c\u9664a\u5f97
\uff081/a)x=A^(-1)(1 1........1)^(-1)
\u79ef\uff081/a 1/a........1/a)=A^(-1)(1 1........1)^(-1)
\u6240\u4ee5A^-1\u7684\u6bcf\u884c\u5143\u7d20\u4e4b\u548c\u4e3a1/a
\u8bc1\u6bd5
\u8bc1\u660e: \u8bbe x=(1,1,...,1)^T.
\u7531\u5df2\u77e5A\u7684\u6bcf\u4e00\u884c\u5143\u7d20\u4e4b\u548c\u4e3ac
\u6240\u4ee5 Ax = (c,c,...,c)^T = cx.
\u6240\u4ee5 A^-1Ax = cA^-1x
\u5373 x = cA^-1x
\u6240\u4ee5 A^-1x = (1/c)x.
--\u6ce8: \u56e0\u4e3aA\u53ef\u9006, \u6545c\u22600
\u6240\u4ee5A^-1\u7684\u6bcf\u4e00\u884c\u5143\u7d20\u4e4b\u548c\u4e3a 1/c.
由A^(-1)·A = E, 有i ≠ j时∑{1 ≤ k ≤ n} b[i,k]·a[k,j] = 0, i = j时∑{1 ≤ k ≤ n} b[i,k]·a[k,j] = 1.
因此1 = ∑{1 ≤ j ≤ n} ∑{1 ≤ k ≤ n} b[i,k]·a[k,j] = ∑{1 ≤ k, j ≤ n} b[i,k]·a[k,j]
= ∑{1 ≤ k ≤ n} ∑{1 ≤ j ≤ n} b[i,k]·a[k,j] = ∑{1 ≤ k ≤ n} b[i,k]·∑{1 ≤ j ≤ n} a[k,j].
而A的各行元素之和均为a ≠ 0, 即∑{1 ≤ j ≤ n} a[k,j] = a对任意1 ≤ k ≤ n成立.
代入得1 = ∑{1 ≤ k ≤ n} b[i,k]·a, 即1/a = ∑{1 ≤ k ≤ n} b[i,k]对任意1 ≤ i ≤ n成立.
也即A^(-1)的各行元素之和均为1/a = a^(-1).
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