Kunze Linear Algebra · Problem 5.4.12

Determinants

Mathematics — Methods · determinant-left-right-multiplication

Problem

If VV is the vector space of n×nn \times n matrices over FF and BB is a fixed n×nn \times n matrix over FF, let LBL_B and RBR_B be the linear operators on VV defined by LB(A)=BAL_B(A) = BA and RB(A)=ABR_B(A) = AB. Show that ((a)) det⁡LB=(det⁡B)n\det L_B = (\det B)^n; ((b)) det⁡RB=(det⁡B)n\det R_B = (\det B)^n.

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