Find the eigenvalues and eigenvectors of the matrix (a) Use the eigenvalues to compute the determinant of

Question:

Find the eigenvalues and eigenvectors of the matrix

(a) Use the eigenvalues to compute the determinant of A.
(b) Is A positive definite? Why or why not?
(c) Find an orthonormal eigenvector basis of R3 determined by A or explain why none exists.
(d) Write out the spectral factorization of A if possible.
(e) Use orthogonality to write the vector (1, 0, 0)T as a linear combination of eigenvectors of A.