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The conclusion you indicate is obtained as the main result in the following paper, but with an apparently stronger hypothesis: (EDIT: Not stronger at all, actually - just realized you're assuming the map is linear.)

Determinant preserving maps on matrix algebras

Gregor Dolinar and Peter Semrl

Linear Algebra and its Applications Volume 348, Issues 1-3, 15 June 2002, Pages 189-192

Let $M_n$ be the algebra of all $n\times n$ complex matrices. If $\phi:M_n→M_n$ is a surjective mapping satisfying $\det(A+\lambda B)=\det(\phi(A)+\lambda\phi(B))$ then either $\phi$ is of the form $\phi(A)=MAN$ or $\phi$ is of the form $\phi(A)=MA^TN$ where $M,N$ are nonsingular matrices with $\det(MN)=1$.

The conclusion you indicate is obtained as the main result in the following paper, but with an apparently stronger hypothesis: (EDIT: Not stronger at all, actually - just realized you're assuming the map is linear.)

Determinant preserving maps on matrix algebras

Gregor Dolinar and Peter Semrl

Linear Algebra and its Applications Volume 348, Issues 1-3, 15 June 2002, Pages 189-192, DOI: 10.1016/S0024-3795(01)00578-X

Let $M_n$ be the algebra of all $n\times n$ complex matrices. If $\phi:M_n→M_n$ is a surjective mapping satisfying $\det(A+\lambda B)=\det(\phi(A)+\lambda\phi(B))$ then either $\phi$ is of the form $\phi(A)=MAN$ or $\phi$ is of the form $\phi(A)=MA^TN$ where $M,N$ are nonsingular matrices with $\det(MN)=1$.

The conclusion you indicate is obtained as the main result in the following paper, but with an apparently stronger hypothesis: (EDIT: Not stronger at all, actually - just realized you're assuming the map is linear.)

Determinant preserving maps on matrix algebras

Gregor Dolinar and Peter Semrl

Linear Algebra and its Applications Volume 348, Issues 1-3, 15 June 2002, Pages 189-192

Let Mn be the algebra of all n×n complex matrices. If φ:Mn→Mn is a surjective mapping satisfying det(A+λB)=det(φ(A)+λφ(B)) then either φ is of the form φ(A)=MAN or φ is of the form φ(A)=MAtN where M,N are nonsingular matrices with det(MN)=1.

The conclusion you indicate is obtained as the main result in the following paper, but with an apparently stronger hypothesis: (EDIT: Not stronger at all, actually - just realized you're assuming the map is linear.)

Determinant preserving maps on matrix algebras

Gregor Dolinar and Peter Semrl

Linear Algebra and its Applications Volume 348, Issues 1-3, 15 June 2002, Pages 189-192

Let $M_n$ be the algebra of all $n\times n$ complex matrices. If $\phi:M_n→M_n$ is a surjective mapping satisfying $\det(A+\lambda B)=\det(\phi(A)+\lambda\phi(B))$ then either $\phi$ is of the form $\phi(A)=MAN$ or $\phi$ is of the form $\phi(A)=MA^TN$ where $M,N$ are nonsingular matrices with $\det(MN)=1$.

The conclusion you indicate is obtained as the main result in the following paper, but with an apparently stronger hypothesis: (EDIT: Not stronger at all, actually - just realized you're assuming the map is linear.)

Determinant preserving maps on matrix algebras

Gregor Dolinar and Peter Semrl

Linear Algebra and its Applications Volume 348, Issues 1-3, 15 June 2002, Pages 189-192

Let Mn be the algebra of all n×n complex matrices. If φ:Mn→Mn is a surjective mapping satisfying det(A+λB)=det(φ(A)+λφ(B)) then either φ is of the form φ(A)=MAN or φ is of the form φ(A)=MAtN where M,N are nonsingular matrices with det(MN)=1.

The conclusion you indicate is obtained as the main result in the following paper, but with an apparently stronger hypothesis: (EDIT: Not stronger at all, actually - just realized you're assuming the map is linear.)

Determinant preserving maps on matrix algebras

Gregor Dolinar and Peter Semrl

Linear Algebra and its Applications Volume 348, Issues 1-3, 15 June 2002, Pages 189-192

Let Mn be the algebra of all n×n complex matrices. If φ:Mn→Mn is a surjective mapping satisfying det(A+λB)=det(φ(A)+λφ(B)) then either φ is of the form φ(A)=MAN or φ is of the form φ(A)=MAtN where M,N are nonsingular matrices with det(MN)=1.