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Vector space of matrices, determiant preserving transform

Source: 2019 South Korea USMC P8

August 13, 2020

vector spaceMatriceslinear algebracollege contestsmatrix

Problem Statement

M_n(\mathbb{C})

is the vector space of all complex

n\times n

matrices. Given a linear map

T:M_n(\mathbb{C})\to M_n(\mathbb{C})

s.t.

\det (A)=\det(T(A))

for every

A\in M_n(\mathbb{C})

. (1) If

T(A)

is the zero matrix, then show that

A

is also the zero matrix. (2) Prove that

\text{rank} (A)=\text{rank} (T(A))

for any

A\in M_n(\mathbb{C})