Problem 34. A matrix A € Mnxn (R) is called skew-symmetric if At = -A.(a) Prove that if A is skew-symmetric and n is odd, then A is not invertible.(b) Find an example of invertible skew-symmetric matrix.

34. (a) A matrix A∈Mn×nℝ is skew symmetric and n is odd. We have to prove that A is not invertible.…

Answered: Problem 34. A matrix A € Mnxn (R) is…

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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Which of the following is true? (1) Either the row vectors or the column vectors of a square matrix are linearly independent. (11) A matrix with linearly independent row vectors and linearly independent column vectors is square. (1II) Adding one additional column to a matrix increases its rank(or lead term) by one. (IV) The nullity of a square matrix with linearly dependent rows is at least one. (V) If A is square and AX = b is inconsistent for some vector b, then the nullity of A is zero.
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 . You may refer to Theorem 1.5.2 or Theorem 2.2.2, but be clear and complete in your argument.
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3. Consider the matrix A -1 4 -3 4 -3 1 -2 0 3 (a) Find matrices P and D that give a diagonalization A = P-¹DP. (b) Is A positive definite, or not? Give a reason for your answer.
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Let A be a 3x3 invertible matrix. The reduced row-echelon form of A is obtained by performing the following three elementary row operations in order: (1) First multiply row 1 by 1. (ii) Then add 3 times row 3 to row 2. (iii) Then finally interchange rows 3 and 2. Then what is det(A)? det(A) = 0

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Problem 34. A matrix A € Mnxn (R) is called skew-symmetric if At = -A. (a) Prove that if A is skew-symmetric and n is odd, then A is not invertible. (b) Find an example of invertible skew-symmetric matrix.