(5)Complex numbers have a representation as two by two matrices with real entries.In this problem you will be asked to show that the matrix operation indeed reproduces thecomplex number operations. For any complex number z assign a matrix M, as follows:z = a + ib→ M₂ =For example we have M1+2i(a) Show that M₂₁ + M22and real number c.=1or M3iMz1+z2 and CM₂a[0-3]- [0= Mez, for any complex numbers z, 21, 22=(b) Show that MzMz2 = Mz122, for any complex numbers 21, 22.(c) Show that M₁/2 = M¹, for any nonzero complex number z.(d) Find an expression for det (M₂) and M in terms of operations of complex numbers.(e) Find the eigenvalues and corresponding eigenvectors of M₂ for z = a + ib. (Note thatyou can only use the techniques used in the section 4.1).

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(5) Complex numbers have a representation as two by two matrices with real entries. In this problem you will be asked to show that the matrix operation indeed reproduces the complex number operations. For any complex number z assign a matrix M, as follows: z = a + ib→ M₂ = For example we have M1+2i = (a) Show that M2₁ + M22 and real number c. 1 or M31 Mz1+z2 and CM₂ a = - [ = Mcz, for any complex numbers z, 21, 22 [0-3] 0 (b) Show that MzMz2 = Mz1z2, for any complex numbers 21, 22. 2122 (c) Show that M₁/2 = M¹, for any nonzero complex number z. (d) Find an expression for det (M₂) and MT in terms of operations of complex numbers. (e) Find the eigenvalues and corresponding eigenvectors of M₂ for z = a + ib. (Note that you can only use the techniques used in the section 4.1).

(5) Complex numbers have a representation as two by two matrices with real entries. In this problem you will be asked to show that the matrix operation indeed reproduces the complex number operations. For any complex number z assign a matrix M, as follows: z = a + ib→ M₂ = For example we have M1+2i = (a) Show that M2₁ + M22 and real number c. 1 or M31 Mz1+z2 and CM₂ a = - [ = Mcz, for any complex numbers z, 21, 22 [0-3] 0 (b) Show that MzMz2 = Mz1z2, for any complex numbers 21, 22. 2122 (c) Show that M₁/2 = M¹, for any nonzero complex number z. (d) Find an expression for det (M₂) and MT in terms of operations of complex numbers. (e) Find the eigenvalues and corresponding eigenvectors of M₂ for z = a + ib. (Note that you can only use the techniques used in the section 4.1).

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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Question

(5)
Complex numbers have a representation as two by two matrices with real entries.
In this problem you will be asked to show that the matrix operation indeed reproduces the
complex number operations. For any complex number z assign a matrix M, as follows:
z = a + ib→ M₂ =
For example we have M1+2i
(a) Show that M₂₁ + M22
and real number c.
=
1
or M3i
Mz1+z2 and CM₂
a
[0-3]
- [
0
= Mez, for any complex numbers z, 21, 22
=
(b) Show that MzMz2 = Mz122, for any complex numbers 21, 22.
(c) Show that M₁/2 = M¹, for any nonzero complex number z.
(d) Find an expression for det (M₂) and M in terms of operations of complex numbers.
(e) Find the eigenvalues and corresponding eigenvectors of M₂ for z = a + ib. (Note that
you can only use the techniques used in the section 4.1).

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