n维列向量组a1...an线性无关 A是n阶方阵 如果Aa1...Aan线性相关 则|A|...答:因为 n维列向量组a1...an线性无关 所以 |a1,...,an| ≠ 0 同理 |Aa1,...,Aan| =0 而 A(a1,...,an) = (Aa1,...,Aan)所以 |A| |a1,...,an| = |Aa1,...,Aan| = 0.故有 |A| = 0.
设n阶方阵A不可逆,n维列向量组a1,a2,...an线性无关,试证明:向量组Aa1...答:R(Aa1,Aa2...Aan)= R(A(a1,a2,...,an))= R(A) ... (a1,a2,...,an) 可逆 < n ... A不可逆 所以 Aa1,Aa2...Aan 线性相关