Let A_n be the n x n matrix with 8 on the main diagonal and 3 elsewhere. For exa

Question

Let A_n be the n x n matrix with 8 on the main diagonal and 3 elsewhere. For example, A_3 = [8 3 3 3 8 3 3 3 8] a) For n = 3, 4, 5 Use Matlab pre-programmed matrices (eye, ones, zeros) and matrix operations, to efficiently input A_n. Compute A_n^-1 using a Matlab command and display the result with rational entries. b) Propose a general form for A_n^-1, expressed in terms on n. c) Check your theory for n = 6. Consider the matrix D = A_4 +[4, -2, 1, 5, 3, 8, 2, -1; 6, 8, 9, 2; 2, 3, -1, 0]. (with A_4 as in problem 3). Compute the following five determinants and comment what general properties of determinants your computations at points b-e illustrate: a) det(D); (b) det(D^T) where T stands for transposed; (c)det(D^2); (d) det(3 D); (e) det (D^-1).

Explanation / Answer

for 3rd problem
a)1.function out = identity(n)
out(n,n) = n;
out(1:n+1:end) = 8;
A=out(n,n);
end
2.B=inv(A)
a=rats(B)

b)B=inv(A)
c)function out = identity(6)
out(6,6)=6;
out(1:n+1:end) =8;
A=out(6,6);
end

for problem4
D=[12,2,5,9;7,16,6,3;10,12,17,6;6,7,3,8]

a)det(D)
d=det(D)

b)det(D transpose)
A=transpose(D)
d=det(A)

c)det(D2)
C=D^2
d=det(c)

d)det(3D)
d=det(D)
C=3.*d

e)det(Dinverse)
c=inv(D)
d=det(c)

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