Reference no: EM132590388

Engineering Mathematics Questions -

Q1. Find the characteristic polynomial and the eigenvalues for the matrix

Q2. This problem establishes a special case of the Cayley-Hamilton theorem.

a) Prove that if B is a (3 x 3) matrix and if Bx = 0 for every x in R³, then B is the zero matrix. [Hint: Consider Be₁, Be₂, and Be₃.]

b) Suppose that λ₁, λ₂, and λ₃ are the eigenvalues of a (3 x 3) matrix A, and suppose that u₁, u₂, and u₃ are corresponding eigenvectors. Prove that if {u₁, u₂, u₃} is a linearly independent set, and if p(t) is the characteristic polynomial of A, then p(A) is the zero matrix.

Q3. For each eigenvalue λ of F, as in problem 1, find a basis for the associated eigenspace E_λ. Determine the algebraic and geometric multiplicities of λ.

Q4. Let P be an idempotent matrix. Show that the only eigenvalues of P are λ = 0 and λ = 1.

Q5. Let u be a unit vector of Rⁿ. Show that the matrix P = uu^T is an idempotent matrix.

Reference: Introduction to Linear Algebra 5ed, by Johnson et. al.