3. Let \( A, B \in M_{n \times n}(\mathbb{C}) \) be the space of square matrices with complex entries and let \( O \in M_{n \times n}(\mathbb{C}) \) be the zero matrix. In the following, remember that a proof is a justification for all \( n \), not just a few specific examples or values of \( n \).(a) Prove that if \( A \) is similar to \( B \) (so there is an invertible \( Q \) such that \( A = Q^{-1}BQ \)), then \( \det(A) = \det(B) \).(b) Suppose there is a positive integer \( k \) such that \( A^k = O \). Prove that \( A \) is not invertible.(c) Prove that if \( A^T = -A \) and \( n \) is odd, then \( A \) is not invertible.

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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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3. Assume A, B are an n x n invertible matrices and c, c‡0 is a scalar, prove the following statements: Hint: To show a matrix is an inverse of another you will need to show left and right multiplication holds! Rely on the following definition (from Section 2.2) for invertible matrices in your proofs: An n x n matrix A is said to be invertible if there is an n x n matrix C such that CA = I and AC = I. (a) (4¹)¹ = A (c) (AB)¹ =B¹A-¹ (d) (4²) ¹ = (^-¹)" 1 (b) (CA)-¹-A¹ =
2. a c) Find all the matrices A = [] with real entries such that A² = [ 2] detail work. Show your
Find the examples that satisfy the statement in the given items and show some work how they are correct examples. (a) A 4 × 4 matrices A that is not digonalizable.(b) Two matrices A and B that have the same characteristic polynomial but are not similar.(c) Two diagonal matrices D1 and D2 that are similar but they are not equal.(d) Two similar matrices A and B such that A is orthogonally digonalizable but B is not orthogonally diagonalizable.
5. Show that a complex square matrix is unitary if and only if it is unitarily equivalent to a complex diagonal matrix whose diagonal entries all have absolute value one.
Which of the following six sets are subrings of M(R)? Which ones have an identity? 0 0 (a) All matrices of the form (8 5) with reQ. (b) All matrices of the form (c) All matrices of the form (d) All matrices of the form (a b) 0 Ca a a with a, b, cЄZ. b with a, b, c = R. 0 with a Є R. 0 with a ЄR. (e) All matrices of the form a (69) (f) All matrices of the form 0 (a) with a ЄR. 0
[9 3 0] b) Given that: A=2 5 6, B-5 li 2 5] [2 -4 1] 4 2 3] Show that matrix multiplication is not commutative i.e., AB # BA
6. Prove a shortened version of the fact from the prior homework using theory of Ch. 3 [if you are utilizing theorems or prior results, cite them!]. Let A and B be square matrices. If AB = I, then A and B are both invertible.
Regarded in M3(F5), where F5 is the field of five elements, put the matrix [0 2 4] 420 003 A = into canonical form for equivalence using only row operations. (Hint: you may wish to use that 2-¹ = 3, 3-¹ = 2, 4¯¹ = 4 when working in F5.)
Suppose Ax = b and Ay = b* are both solvable. Then Az = b + b* is solvable. What is z? This translates into: If b and b* are in the column space C(A), then b + b* is in C(A).
3) We know, by theorem, that if A is a square matrix then the following statements are equivalent: (a) A is invertible. (b) AX=0 has only the trivial solution. (c) The RREF of A is I. (d) A is expressible as a product of elementary matrices. 24 Consider the following matrix, A = [34] 01 Verify that all 4 conditions in the theorem are either all true or all false for the given matrix. That is, if A is invertible, show that the 3 other conditions are true, as well. If it is not, show that they are all false.
Find a non-zero 2 x 2 matrix such that 1 3 4 -12 Commun [88]
Provide complete arguments/proofs for the following. a) A is invertible iff there exists a matrix B so that AB = BA = I. It is simple to show that: (i) If A is invertible and AB = I, then BA = I as well and B is the unique such matrix. (ii) If A is invertible and BA = I, then AB = I as well and B is the unique such matrix. This shows that if A is invertible, then there is a unique matrix B such that AB = I or BA = I. Call this unique matrix A-1. The goal here is to show that the assumption “A is invertible” is not needed in (i) or (ii). Prove: Let A and B be square matrices with AB = I. Show that A is invertible and hence B = A-1. b) Prove: NS(A) = NS(AT A) for any matrix A. c) Prove: If A and B are m x n matrices such that Ax = Bx for all x∈ Rn, then A = B.

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