## Question

###### Let A aud B be n following identity:matrices ad B; be the ith column 0l B for = [We want t0 prove thedet([ABB,] +det([Bi ABzBa]~det([BABa] = tr(A) det (BIF You S prore the above dlirectly; do it . Otherwise follow the following steps;Show that for any fixed B the mapping T : R"x" _ R defined byT(A) = det([ABBal det([B AB,B,]det( [B1AB,]linear transformation; Let {E;j:i.j=1,2 "} be the standard basis of R"xn where E,j is the n X n matrix with in the (i. j)-entry and with 0 in

Let A aud B be n following identity: matrices ad B; be the ith column 0l B for = [ We want t0 prove the det([AB B,] +det([Bi ABz Ba] ~det([B ABa] = tr(A) det (B IF You S prore the above dlirectly; do it . Otherwise follow the following steps; Show that for any fixed B the mapping T : R"x" _ R defined by T(A) = det([AB Bal det([B AB, B,] det( [B1 AB,] linear transformation; Let {E;j:i.j=1,2 "} be the standard basis of R"xn where E,j is the n X n matrix with in the (i. j)-entry and with 0 in other entries_ Show that T(E;;) tr(E;j) det(B). Show that T(A) tr( A) dlet ( B).

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