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please do both 1 & 2 () There is interesting relationship2 between a matrix and its characteristic equation that...
3. (a) For the following matrix A, compute the characteristic polynomial C(A) = det(A ?): A-1...
3. (a) For the following matrix A, compute the characteristic polynomial C(A) = det(A ?): A-1 1 (b) Find all eigenvalues of A, using the following additional information: This miatrix has exactly 2 eigenvalues. We denote these ??,A2, where ?1 < ?2. . Each Xi is an integer, and satisfies-2 < ?? 2. (c) Given an eigenvalue ?? of A, we define the corresponding eigenspace to be the nullspace of A-?,I; note that this consists of all eigenvectors corresponding to...
Problem 2. (a) Let A be a 4 x 4 matrix with characteristic polynomial p(t) =...
Problem 2. (a) Let A be a 4 x 4 matrix with characteristic polynomial p(t) = +-12+} Find the trace and determinant of A. 2 e: tr(4) and det(A) = 0 12: tr(A) = 0 and det(A) 2 3 2 T: tr(A) = 0 and det(A) 3 : None of the other answers 01 OW
3. Let A be the matrix 1 -2 (a) What is the characteristic polynomial for A?...
3. Let A be the matrix 1 -2 (a) What is the characteristic polynomial for A? (b) What are the eigenvalues of A? (c) What are the eigenvectors of A?
Suppose that a matrix A has the characteristic polynomial (1+1)3 (a 1 + 12 + b)...
Suppose that a matrix A has the characteristic polynomial (1+1)3 (a 1 + 12 + b) for some a, b € R. If the trace of A is 7 and the determinant of A is -24, find all eigenvalues of A. (a) Enter the eigenvalues as a list in increasing order, including any repetitions. For example, if they are 1,1,0 you would enter 0,1,1: (b) Hence determine a: Number (c) and b: Number
The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation...
The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of the matrix shown below is as follows. 1-3 A = 12 - 61 + 11 = 0 and by the theorem you have A2 - 64 + 1112 = 0 2 5 Demonstrate the Cayley-Hamilton Theorem for the matrix A given below. 0 5 -1 -1 3 1 0 0 1 STEP 1: Find and expand the characteristic equation. STEP 2: Compute the...
8. Let A be an nxn matrix with distinct n eigenvalues X1, 2... (a) What is...
8. Let A be an nxn matrix with distinct n eigenvalues X1, 2... (a) What is the determinant of A. (b) If a 2 x 2 matrix satisfies tr(AP) = 5, tr(A) = 3, then find det(A). (The trace of a square matrix A, denoted by tr(A), is the sum of the elements on the main diagonal of A.
The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation...
The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of the matrix shown below is as follows. --1:: 22 - 61 + 11 = 0 and by the theorem you have 42 - 64 + 1112 = 0 Demonstrate the Cayley-Hamilton Theorem for the matrix A given below. 03 1 A = -1 5 1 0 0 -1 STEP 1: Find and expand the characteristic equation. STEP 2: Compute the required powers of...
Differention Equations - Can someone answer the checked numbers please? Determinants 659 is the characteristic equation of A with λ replaced by /L we can multiply by A-1 to get o get Now solve fo...
Differention Equations - Can someone answer the checked
numbers please?
Determinants 659 is the characteristic equation of A with λ replaced by /L we can multiply by A-1 to get o get Now solve for A1, noting that ao- det A0 The matrix A-0 22 has characteristic equation 0 0 2 2-A)P-8-12A +62- 0, so 8A1-12+6A -A, r 8A1-12 Hence we need only divide by 8 after computing 6A+. 23 1 4 12 10 4 -64 EXERCISES 1. Find AB,...
= ) 1 5 # 4 [10 points] Recall the real symmetric 2 by 2 matrix...
= ) 1 5 # 4 [10 points] Recall the real symmetric 2 by 2 matrix B of # 3. 5 1 # 4 a) From the matrix fact sheet, what are the eigenvalues of B and its characteristic polynomial ? # 4 b) Compute the trace of B, the sum of diagonal terms of the matrix B in three ways. 1- directly, 2- via eigenvalues, 3-via characteristic polynomial # 4 c) Compute the determinant of B, ad — bc,...
1. For a polynomial p(1) = cktk + Ck-14k-1 +...+ci+co, and an n x n matrix...
1. For a polynomial p(1) = cktk + Ck-14k-1 +...+ci+co, and an n x n matrix A, we define p(A) = CkAk + Ck-1 Ak-1 + ... +CjA + col. Let A be an n x n diagonalizable matrix with characteristic polynomial PA(1) = (1-2)*(1-3)n-k where 1 <k<n - 1. In other words, let A be an n x n diagonal- izable matrix that has only 2 and 3 as eigenvalues. Explain what is wrong with the following false "proof...
3. (a) For the following matrix A, compute the characteristic polynomial C(A) = det(A ?): A-1 1 (b) Find all eigenvalues of A, using the following additional information: This miatrix has exactly 2 eigenvalues. We denote these ??,A2, where ?1 < ?2. . Each Xi is an integer, and satisfies-2 < ?? 2. (c) Given an eigenvalue ?? of A, we define the corresponding eigenspace to be the nullspace of A-?,I; note that this consists of all eigenvectors corresponding to...
Problem 2. (a) Let A be a 4 x 4 matrix with characteristic polynomial p(t) = +-12+} Find the trace and determinant of A. 2 e: tr(4) and det(A) = 0 12: tr(A) = 0 and det(A) 2 3 2 T: tr(A) = 0 and det(A) 3 : None of the other answers 01 OW
3. Let A be the matrix 1 -2 (a) What is the characteristic polynomial for A? (b) What are the eigenvalues of A? (c) What are the eigenvectors of A?
Suppose that a matrix A has the characteristic polynomial (1+1)3 (a 1 + 12 + b) for some a, b € R. If the trace of A is 7 and the determinant of A is -24, find all eigenvalues of A. (a) Enter the eigenvalues as a list in increasing order, including any repetitions. For example, if they are 1,1,0 you would enter 0,1,1: (b) Hence determine a: Number (c) and b: Number
The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of the matrix shown below is as follows. 1-3 A = 12 - 61 + 11 = 0 and by the theorem you have A2 - 64 + 1112 = 0 2 5 Demonstrate the Cayley-Hamilton Theorem for the matrix A given below. 0 5 -1 -1 3 1 0 0 1 STEP 1: Find and expand the characteristic equation. STEP 2: Compute the...
8. Let A be an nxn matrix with distinct n eigenvalues X1, 2... (a) What is the determinant of A. (b) If a 2 x 2 matrix satisfies tr(AP) = 5, tr(A) = 3, then find det(A). (The trace of a square matrix A, denoted by tr(A), is the sum of the elements on the main diagonal of A.
The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of the matrix shown below is as follows. --1:: 22 - 61 + 11 = 0 and by the theorem you have 42 - 64 + 1112 = 0 Demonstrate the Cayley-Hamilton Theorem for the matrix A given below. 03 1 A = -1 5 1 0 0 -1 STEP 1: Find and expand the characteristic equation. STEP 2: Compute the required powers of...
Differention Equations - Can someone answer the checked
numbers please?
Determinants 659 is the characteristic equation of A with λ replaced by /L we can multiply by A-1 to get o get Now solve for A1, noting that ao- det A0 The matrix A-0 22 has characteristic equation 0 0 2 2-A)P-8-12A +62- 0, so 8A1-12+6A -A, r 8A1-12 Hence we need only divide by 8 after computing 6A+. 23 1 4 12 10 4 -64 EXERCISES 1. Find AB,...
= ) 1 5 # 4 [10 points] Recall the real symmetric 2 by 2 matrix B of # 3. 5 1 # 4 a) From the matrix fact sheet, what are the eigenvalues of B and its characteristic polynomial ? # 4 b) Compute the trace of B, the sum of diagonal terms of the matrix B in three ways. 1- directly, 2- via eigenvalues, 3-via characteristic polynomial # 4 c) Compute the determinant of B, ad — bc,...
1. For a polynomial p(1) = cktk + Ck-14k-1 +...+ci+co, and an n x n matrix A, we define p(A) = CkAk + Ck-1 Ak-1 + ... +CjA + col. Let A be an n x n diagonalizable matrix with characteristic polynomial PA(1) = (1-2)*(1-3)n-k where 1 <k<n - 1. In other words, let A be an n x n diagonal- izable matrix that has only 2 and 3 as eigenvalues. Explain what is wrong with the following false "proof...
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