M is a symmetric 4 x 4 matrix with the property that Ms = 2s for...

 

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M is a symmetric 4 x 4 matrix with the property that Ms = 2s for all s S where S= 2 R 1+2+x3=0, x1x3=0 and x4 = 0 23 A and Mq=3q for all qQwhere 22 Q ER 1-2x2 = 0 and x3 = 0 23 Moreover, det (M) = -18. (i) Find a basis B of S and a basis B of Q. (ii) Find the eigenvalues of M and state the algebraic and geometric multiplicity of each eigen- value. (iii) Find an invertible matrix P and a diagonal matrix D such that M = PDP-1 M is a symmetric 4 x 4 matrix with the property that Ms = 2s for all s S where S= 2 R 1+2+x3=0, x1x3=0 and x4 = 0 23 A and Mq=3q for all qQwhere 22 Q ER 1-2x2 = 0 and x3 = 0 23 Moreover, det (M) = -18. (i) Find a basis B of S and a basis B of Q. (ii) Find the eigenvalues of M and state the algebraic and geometric multiplicity of each eigen- value. (iii) Find an invertible matrix P and a diagonal matrix D such that M = PDP-1