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How to show $\det(AB) =\det(A)\det(B)$

How would I prove that

$$\det{AB} = \det{BA} = \det{A}\det{B}$$

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marked as duplicate by userNaN, Chris Taylor, draks ..., Zhen Lin, Marc van Leeuwen Oct 9 '12 at 8:56

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What definition of the determinant are you using? –  Qiaochu Yuan Oct 9 '12 at 8:36

1 Answer 1

From Wikipedia: This property is a consequence of the characterization [given above] of the determinant as the unique $n$-linear alternating function of the columns with value $1$ on the identity matrix, since the function $M_n(K)\to K$ that maps $M\mapsto\det(AM)$ can easily be seen to be $n$-linear and alternating in the columns of $M$, and takes the value $\det(A)$ at the identity.

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