Let A be a 3 by 3 matrix, with eigenvalues
&lgr;[1]=-2; &lgr;[2]=0; &lgr;[3]=5;
(a) Compute tr(A) and det(A).
tr[A]=Apply[Plus,Table[&lgr;[i],{i,3}]]
det[A]=Apply[Times,Table[&lgr;[i],{i,3}]]
(b) Is A invertible? Explain your answer.
No, since det[A]=0.
(c) Is A diagonalizable? Explain your answer.
Yes, since the eigenvalues are all distinct.
(d) Compute tr(A^3) and det(A^3).
tr[A^3]=Apply[Plus,Table[&lgr;[i]^3,{i,3}]]
det[A^3]=Apply[Times,Table[&lgr;[i]^3,{i,3}]]