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1. The determinant of a 3 x 3 matrix [A] is defined as A1A2A3 det[A]=[A1A22A3|=e A2 A32
(a) Show that ejk A, Ajq A. = epqr det[A].
(b) From (a), show that det[A]=A,, and det [A]T=det [A]
(c) Let [A] and [B] be 3 x 3 matrices. Using (a) and (b), show that det([A][B])=det([A])det([B])
(d) Let a, b, c, u, v, and w be three-dimensional vectors. In the class, we have shown abc= ea.b;c,=det[[a][b][c]]. Use the results of (a) and (c) to show (axbc)(uxvw)=|bu bv bw
c.u c.v
cw
(e) Using these results, obtain the following identities relating the permutation Sip Gig "s O ja Oj N Or Or
(f) Use (e) to show that eukepak =&i,&;