q7 provide handwritten sol for this ques asap 7Read the following carefully, then answer the questions below.One aspect of matrices we did not explore much in this course is matrices withcomplex number entries. It ends up that many of the same ideas that we studied for(real number) matrices have analogues when we allow complex numbers.Suppose that A is a matrix with complex entries ajj.We define the complex conjugate of A, written A, to be the matrix whose entriesare āij (i.e. we conjugate each entry of A).Furthermore, we define the the conjugate transpose of A, written A+, to be thematrix A* = A', i.e. the transpose of A.First, there are some natural algebraic facts that can be verified in a straightforwardfashion:If A and B are two square matrices with complex entries and z is any complexnumber, then the following are true:1. (A*)+ = A2. (kA)+ = KA+3. (A+ B)+ = A+ +B+4. (AB)+ = B+A+We now define a special kind of complex matrix:8 A square matrix A with complex entries is called awesome if A+ = A-1.7.1Determine whether the matrix B =is awesome or not.Suppose that At is a 2 x 2 matrix and that At = -A. Show7.2that iA s awesome.: If A and B are two awesome matrices of the same7.3size, then AB is awesome.TheoremDefinition

7.1 To determine whether the matrix B is awesome or not. Where B=i2−3232−i2and B is awesome if…

Answered: Determine whether the matrix B = is…

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Determine whether the matrix B = is awesome or not. %3D

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter2: Matrices

Section2.2: Properties Of Matrrix Operations

Problem 79E

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q7 provide handwritten sol for this ques asap

7
Read the following carefully, then answer the questions below.
One aspect of matrices we did not explore much in this course is matrices with
complex number entries. It ends up that many of the same ideas that we studied for
(real number) matrices have analogues when we allow complex numbers.
Suppose that A is a matrix with complex entries ajj.
We define the complex conjugate of A, written A, to be the matrix whose entries
are āij (i.e. we conjugate each entry of A).
Furthermore, we define the the conjugate transpose of A, written A+, to be the
matrix A* = A', i.e. the transpose of A.
First, there are some natural algebraic facts that can be verified in a straightforward
fashion:
If A and B are two square matrices with complex entries and z is any complex
number, then the following are true:
1. (A*)+ = A
2. (kA)+ = KA+
3. (A+ B)+ = A+ +B+
4. (AB)+ = B+A+
We now define a special kind of complex matrix:
8 A square matrix A with complex entries is called awesome if A+ = A-1.
7.1
Determine whether the matrix B =
is awesome or not.
Suppose that At is a 2 x 2 matrix and that At = -A. Show
7.2
that iA s awesome.
: If A and B are two awesome matrices of the same
7.3
size, then AB is awesome.
Theorem
Definition

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