Suppose W, X and Y are matrices with the following properties.W has characteristic polynomial A²+3. A + 10.X has characteristic polynomial (A − 1) · (A² + 3 · A).Y has characteristic polynomial A² + 12 - A+ 36.(i.) Which one of the three matrices has no real eigenvalues? (No answer given)(ii.) Calculate the quantity trace(W) - det (W).(iii.) Calculate rank(X) and rank(Y).rank(X) =rank (Y) =(iv.) Which one of the three matrices describes a linear transformation with non-trivial fixedpoints? (No answer given)

As you posted multiple subparts I have to solve only first three subparts.

Answered: Suppose W, X and Y are matrices with…

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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What is the definition of the eigenvalues λλ of a matrix A.A? (I.I refers to the identity matrix.) a. The roots of det (λA − I) b. The roots of det (λA + I) c. The roots of det (A + λI) d. The roots of det (A − λI)
3. Find the eigenvalues of the following matrices and then find a basis for each eigenspaces. 4 [213] (a) A (b) A = (c) A (d) B = || 6 2√3 }} 2√3 7 (e) C = 1 1 1 1 −4 -4 2 -2 -2 222
-2 2. Find h in the matrix A = 7 such that the eigenspace for 1 = 7 is one- -1 4 -6 5 7 dimensional. Find h such that the eigenspace for 1=7 is two-dimensional.
solve each set of equations by the method of finding the inverse of the coefficient matrix. Hint: See Example 3. y + z = 4 2x + y z = -1 3x + 2y + 2z =
A X 2) Consider a matrix M = (6 %) in block form, where A and B are square matrices. Prove that 0 B det (M) = det (A) det (B). Suggestion: Induct on the number of rows of A. Use cofactor expansions down the first column. Each of the cofactors appearing is a block matrix with one fewer rows than A.
Which of the following statements are always true? (i) If A is a 3 x 3 diagonalizable matrix then A has 3 distinct eigenvalues. (ii) If the characteristic polynomial of a matrix A is equal to (2 – 3)*(2– 6)°(2 – 7)š then the eigenvalue 2 = 3 has a geometric multiplicity of 4. (iii) If A is similar to B, and B is symmetric, then A' is similar to B. A) (i) and (ii) only (B) (ii) and (iii) only (C) (ii) only (D) none of them (E) (iii) only (F) (i) only G) all of them (H) (i) and (iii) only
1.10 Given any m × n matrices A and B, solve the matrix equation 3(X + ½ A) = 5(X − 2(B). -
Which one of the following is the characteristic equation of the matrix (1 − A)² (5 — A) = 0 (1 – A²)(5 — A) = 0 (1 − A)(5 — X) = 0 (1 − A)(5 — A)² = 0 2 0 7 1 3 0 7 4 ?
Example. Find the eigenvalues and eigenvectors of the following matrix: 0. -2 М— -2i i 2i [3.73] 0. -1
Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. To save time, the eigenvalues are 10 and -9. A = 1 -3 9 -3 93 9 3 1 Enter the matrices P and D below. (Use a comma to separate answers as needed. Type exact answers, using radicals as needed. Do not label the matrices.)
Help me please
Let 1 3 4. A = -5 -3 and B = -4 -6 -3 3 1 3 1 For this problem, you may use the fact that both matrices have the same characteristic polynomial: PA(A) = PB(A) = -(A – 1)(A+2)². (a) Find all eigenvectors of A. (b) Find all eigenvectors of B. (C) Which matrix A or Bis diagonalizable? (d) Diagonalize the matrix stated in (C), i.e., find an invertible matrix P and a diagonal matrix D such that A = PDP B= PDP 1. 1 or %3D %3D

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