Question number 10 and 11 10. (In this problem we show any square matrix is the sum of a symmetric matrix and askew-symmetric matrix).Let A be a square matrix. From this defineand=(d) An example: for A=1 23 4-Bс-(a) Using general properties of the transpose, pg. 36, show B is symmetric, i.e.Bt B.72=1(b) Using general properties of the transpose, pg. 36, show C is skew-symmetric, i.e.Ct=-C.(c) Show B+C = A.(4+(A + A¹)21(4-(A – Aª).11. For two nx n matrices A and B, we define the commutator, denoted [A, B], by[A, B]AB - BA.a) What is [A, I], where I is the n x n identity matrix?b) What is [A, A] for any n x n matrix A ?c) Give an example of two nonzero 2 × 2 matrices A and B such that [A, B] is not thezero matrix.d) Fix an n x n matrix C. Show that [A + B, C] = [A, C] + [B, C]e) [Optional but fun!] Compute out this sumcompute B and C.[A, [B, C]] + [B, [C, A]] + [C, [A, B]].

D. (In this problem we show any square matrix is the sum of a symmetric matrix and a skew-symmetric matrix). Let A be a square matrix. From this define and C=(A-A¹). (a) Using general properties of the transpose, pg. 36, show B is symmetric, i.e. B = B. 1 B = (A + A¹) (b) Using general properties of the transpose, pg. 36, show C is skew-symmetric, i.e. Ct =-C. (c) Show B+C = A. (d) An example: for A = 1 2 34 compute B and C.

D. (In this problem we show any square matrix is the sum of a symmetric matrix and a skew-symmetric matrix). Let A be a square matrix. From this define and C=(A-A¹). (a) Using general properties of the transpose, pg. 36, show B is symmetric, i.e. B = B. 1 B = (A + A¹) (b) Using general properties of the transpose, pg. 36, show C is skew-symmetric, i.e. Ct =-C. (c) Show B+C = A. (d) An example: for A = 1 2 34 compute B and C.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.1: Matrix Operations

Problem 20EQ: Referring to Exercise 19, suppose that the unit cost of distributing the products to stores is the...

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