This key fact can be proven by observing that, for a fixed matrix , both sides of the equation are alternating and multilinear as a function depending on the columns of . Moreover, they both take the value when is the identity matrix. The above-mentioned unique characterization of alternating multilinear maps therefore shows this claim.Alternatively, proves this result using the functoriality of the exterior power.
This key fact can be proven by observing that, for a fixed matrix , both sides of the equation are alternating and multilinear as a function depending on the columns of . Moreover, they both take the value when is the identity matrix. The above-mentioned unique characterization of alternating multilinear maps therefore shows this claim.Alternatively, proves this result using the functoriality of the exterior power.