MathDB

8

Part of 2019 Korea USCM

Problems(1)

Vector space of matrices, determiant preserving transform

Source: 2019 South Korea USMC P8

8/13/2020
Mn(C)M_n(\mathbb{C}) is the vector space of all complex n×nn\times n matrices. Given a linear map T:Mn(C)→Mn(C)T:M_n(\mathbb{C})\to M_n(\mathbb{C}) s.t. det⁡(A)=det⁡(T(A))\det (A)=\det(T(A)) for every A∈Mn(C)A\in M_n(\mathbb{C}). (1) If T(A)T(A) is the zero matrix, then show that AA is also the zero matrix. (2) Prove that rank(A)=rank(T(A))\text{rank} (A)=\text{rank} (T(A)) for any A∈Mn(C)A\in M_n(\mathbb{C}).
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