Problem B6 (40 pts). Consider the following nn matrix, for n3 : B=111111111111111111111111111111111111 (1) If...
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Problem B6 (40 pts). Consider the following nn matrix, for n3 : B=111111111111111111111111111111111111 (1) If we denote the columns of B by b1,,bn, prove that (n3)b1(b2++bn)b1b2b1b3b1bn=2(n2)e1=2(e1+e2)=2(e1+e3)=2(e1+en) where e1,,en are the canonical basis vectors of Rn. (2) Prove that B is invertible and that its inverse A=(aij) is given by a11=2(n2)(n3),ai1=2(n2)12in and aii=2(n2)(n3),2inaji=2(n2)1,2in,j=i (3) Show that the n diagonal nn matrices Di defined such that the diagonal entries of Di are equal the entries (from top down) of the i th column of B form a basis of the space of nn diagonal matrices (matrices with zeros everywhere except possibly on the diagonal). For example, when n=4, we have D1D3=1000010000100001=1000010000100001D2D4=1000010000100001.=1000010000100001
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