设n维列向量a=(a,0,…,0,a)T(a>0)且A=E-aaT,A-1=E+1/a·aaT,则a=________.

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问题 设n维列向量a=(a,0,…,0,a)T(a>0)且A=E-aaT,A-1=E+1/a·aaT,则a=________.

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答案1/2

解析 由于
    AA-1=(E-ααT)(E+1/a·ααT)=E-ααT+1/a·ααT-1/a·ααTααT
    而αTα=2a2,故
    AA-1=E+(-1+1/a-2a)ααT=E.
由此可得-1+1/a-2a=0,解得a=1/2,a=-1(舍去),故a=1/2.
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