4. Let A be the 6*6 diagonal matrix below. For each eigenvalue, compute the multiplicity of...

4. Let A be the 6*6 diagonal matrix below. For each eigenvalue, compute the multiplicity of λ as a root of the characteristic polynomial and compare it to the dimension of the eigenspace Eλ.

(x 0 0 0 0 0 0 x 0 0 0 0 0 0 y 0 0 0 0 0 0 x 0 0 0 0 0 0 z 0 0 0 0 0 0 x)

5. Let A be an 3*3 upper triangular matrix with all diagonal elements equal, such as (3 4 -2 0 3 12 0 0 3) Prove that A is diagonalizable if and only if A is a scalar times the identity matrix.

Expert Solution

Colby Messinger answered 2 years ago

The matrix A=[-2,0,2;0,-4,0;-2,0,-6] has a single eigenvalue=-4 with algebraic multiplicity three. a.find the basis for the...

The matrix A=[-2,0,2;0,-4,0;-2,0,-6] has a single eigenvalue=-4 with algebraic multiplicity three. a.find the basis for the associated eigenspace. b.is the matrix defective? select all that apply. 1. A is not defective because the eigenvalue has algebraic multiplicity 3. 2.A is defective because it has one eigenvalue. 3.A is defective because geometric multiplicity of the eigenvalue is less than the algebraic multiplicity. 4.A is not defective because the eigenvectors are linearly dependent.

Let ? be an eigenvalue of a matrix A. Explain why dim(?) ? 1

5. Compute a basis for each eigenspace of matrix A = | 6 3 −4 −1...

5. Compute a basis for each eigenspace of matrix A = | 6 3 −4 −1 | (a) Diagonalize A. (b) Compute A4 and A−3 . (c) Find a matrix B such that B2 = A. (or compute √ A.)

2. For each 3*3 matrix and each eigenvalue below construct a basis for the eigenspace Eλ....

2. For each 3*3 matrix and each eigenvalue below construct a basis for the eigenspace Eλ. A= (9 42 -30 -4 -25 20 -4 -28 23), λ = 1,3 A= (2 -27 18 0 -7 6 0 -9 8), λ = −1,2 3. Construct a 2×2 matrix with eigenvectors(4 3) and (−3 −2) with eigen-values 2 and −3, respectively.

1. For each matrix A below compute the characteristic polynomial χA(t) and do a direct matrix...

1. For each matrix A below compute the characteristic polynomial χA(t) and do a direct matrix computation to verify that χA(A) = 0. (4 3 -1 1) (2 1 -1 0 3 0 0 -1 2) (3*3 matrix) 2. For each 3*3 matrix and each eigenvalue below construct a basis for the eigenspace Eλ. A= (9 42 -30 -4 -25 20 -4 -28 23),λ = 1,3 A= (2 -27 18 0 -7 6 0 -9 8) , λ = −1,2...

1. For each permutationσ of {1,2,··· ,6} write the permutation matrix M(σ) and compute the determinant...

1. For each permutationσ of {1,2,··· ,6} write the permutation matrix M(σ) and compute the determinant |m(σ)|, which equals sgn(σ). (a) The permutation given by 1 → 2, 2 → 4, 3 → 3, 4 → 1, 5 → 6, 6 → 5. (b) The permutation given by 1 → 5, 2 → 1, 3 → 2, 4 → 6, 5 → 3, 6 → 4.  

Part A: Compute the derivative of ?(?)=(4?^4 + 2?)(?+9)(?−6) Part B: Compute the derivative of ?(?)=...

Part A: Compute the derivative of ?(?)=(4?^4 + 2?)(?+9)(?−6) Part B: Compute the derivative of ?(?)= (9x^2 + 8x +8)(4x^4 + (6/x^2))/x^3 + 8 Part C: Compute the derivative of ?(?)=(15?+3)(17?+13)/(6?+8)(3?+11).

6. Let A = {1, 2, 3, 4} and B = {5, 6, 7}. Let f...

6. Let A = {1, 2, 3, 4} and B = {5, 6, 7}. Let f = {(1, 5),(2, 5),(3, 6),(x, y)} where x ∈ A and y ∈ B are to be determined by you. (a) In how many ways can you pick x ∈ A and y ∈ B such that f is not a function? (b) In how many ways can you pick x ∈ A and y ∈ B such that f : A → B...

Let 3x3 matrix A = -3 0 -4 0 5 0 &nb

Let 3x3 matrix A = -3 0 -4 0 5 0 -4 0 3 a) Find the eigenvalues of A and list their multiplicities. b) Find a basis, Bi, for each eigenspace, E(i). c) If possible, diagonalise matrix A. (i.e find matrices P and D such that Pinv AP = D is diagonal).

Let B equal the matrix below: [{1,0,0},{0,3,2},{2,-2,-1}] (1,0,0 is the first row of the matrix B)...

Let B equal the matrix below: [{1,0,0},{0,3,2},{2,-2,-1}] (1,0,0 is the first row of the matrix B) (0,3,2 is the 2nd row of the matrix B (2,-2,-1 is the third row of the matrix B.) 1) Determine the eigenvalues and associated eigenvectors of B. State both the algebraic and geometric multiplicity of the eigenvalues. 2) The matrix is defective. Nonetheless, find the general solution to the system x’ = Bx. (x is a vector)

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