[Solved] MATH 225 Linear Algebra and Differential Equations2

QUESTIONS:

1)( 40 pts.)Let

be given.

(a)(5 pts.)Find all eigenvalues of A.

(b)(10 pts.)Find a basis for each eigenspace and determine the associated eigenvectors.

(c)(5 pts.)Find an invertible matrix P and a diagonal matrix D such that P¹AP = D.

(d)(10 pts.)Find the inverse of the matrix P in part (c) by using Gaussian elimination method. Verify your answer i.e. show that PP¹ = I = P¹P.

(e)(5 pts.)Calculate A²⁰²⁰.

(f)(5 pts.)Find A¹ by using Cayley-Hamilton Theorem.

• (a)(10 pts.) The characteristic polynomial of a certain 3x3 matrix A is p(x) = x³ 7x² + 5x 9. Use this fact to express adj(A) as a linear combination of A²,A and I.

(b)(10 pts.) If A is an nxn non-singular matrix, show that adj(A) can be

expressed as a linear combination of Aⁿ¹,Aⁿ²,,A,I.

• (10 pts.)The nxn matrix A is said to be idempotent if A² = A. If is an eigenvalue of such a matrix, show that is either 0 or 1. What can be said about a non-singular idempotent matrix?