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Suppose that A+ is a 2 × 2 matrix and that A+ = -A. Show 7.2 that iA is awesome.

Suppose that A+ is a 2 × 2 matrix and that A+ = -A. Show 7.2 that iA is awesome.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.4: Similarity And Diagonalization
Problem 39EQ
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answer 7.2

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 aij. We define the complex conjugate of A, written A, to be the matrix whose entries are āj; (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: 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 A+ is a 2 x 2 matrix and that A+ -A. Show 7.2 that iA is awesome. 7.3 True or False: If A and B are two awesome matrices of the same size, then AB is awesome. Theorem Definition
Transcribed Image Text: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 aij. We define the complex conjugate of A, written A, to be the matrix whose entries are āj; (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: 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 A+ is a 2 x 2 matrix and that A+ -A. Show 7.2 that iA is awesome. 7.3 True or False: If A and B are two awesome matrices of the same size, then AB is awesome. Theorem Definition
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