Reading Question 3.2.3. Which of the following claims are true? Select all that apply.2 1) belongs to GL(2,R).(a) The matrix A =4 2(b) GL(2, R) is an abeliangroup.(c) Every 2 x 2 (real) matrix with determinant 13 belongs to GL2 (R).55(d) If A e GL2(R) satisfies A² = I, then A = ±I.

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Answered: Reading Question 3.2.3. Which of the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question 2. Let A be a matrix in M14 (C) such that • The minimal polynomial of A is (X-A₁)²(X-2) (X-3)²(x-14)³, with A1, A2, A3 and A4 four different complex numbers. rank (A - I) = 11, rank(A - I)² 11, rank (AI)² = 10, rank(A - λ31) = 12, rank(A31)² = 10, rank(A - I) 13. Find the Jordan canonical form of A, and the trace, the determinant and the characteristic polyno- mial of A.
. Compute the determinants of the following matrices by inspection. [2000] 1 0 04 [1 a) A = 00 3 0 2 3 8] b) A = 1 2 7 0 1 -4 00 2 00 LO -3] 1 7 -3]
(b) Consider the following vectors u, v, w, and z u = [9.3 1.6 -0.9], v = [-7.8 5.9 5.3], w = [5.1 0.1 1.3], z = [4.1 8.6 7.9] Find the determinant of the following matrix. u'w u'z z. To check your work, you can also caleulate the following quantity: (u x v) (w x z). This quantity, and the determinant of the above matrix, must always be equal.
1. The group SU (2) is the group of special unitary (2 x 2) matrices, i.e., the unitary (2 x 2) matrices with determinant 1. A representation of the elements of SU (2) is Û (a) = exp(-ã.5) where a Ŝ € R, j = 1,2,3 and 3 = is the spin operator which is proportional to the vector of Pauli matrices & = (₁, 6₂, 63)¹ 02, o 0 1 and 63 = (1₁) 0 2₁ = (₁ 2)₁ 6₂ = (₁ 3²1) 1), ¹) and 1 0 a. Write [Û (a)] as a linear combination of 2 × 2 Pauli matrices to show that Û (a) = cos ½ – i sin ªª. ☎ 22
7
- 5. A linear operator T: V → V is said to be nilpotent if Th 0 for some positive integer k. Similarly, a square matrix A is said to be nilpotent if Ak = 0 for some positive integer k. = (a) Let T: V → V be a linear operator on a finite-dimensional complex inner product space V. Show that if T is normal and nilpotent, then T = 0. (Hint: spectral theorem for normal operators.) (b) Use (a) to show that if a square matrix A is normal and nilpotent, then A = 0. (Hint: apply (a) to T = LA: Cn Cn.) →
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2.2.3 Compute the principal square root of each of the following positive semidefinite matrices. (b) 2 3 3 (d) 2 1 1 2 1 mi 1 1 (f) 2 2 0. 2 2 0 0 1 V3 0 0 V3 1,
7.3 #12 The question is in the picture

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Reading Question 3.2.3. Which of the following claims are true? Select all that apply. 2 1 (a) The matrix A () belongs to GL(2, R). 4 2 (b) GL(2, R) is an abelian group. (c) Every 2 x 2 (real) matrix with determinant 13 belongs to GL2(R). 55 (d) If A E GL2(R) satisfies A² = I, then A = ±I.