Complex numbers have a representation as two by two matrices with real entries. In this problem you will be asked to sho

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Complex numbers have a representation as two by two matriceswith real entries.In this problem you will be asked to show that the matrix operationindeed reproduces thecomplex number operations.For any complex number z assign a matrixM:as follows:
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Complex Numbers Have A Representation As Two By Two Matrices With Real Entries In This Problem You Will Be Asked To Sho 1 (193.52 KiB) Viewed 41 times

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 M₂ as follows: z = a + ib→ M₂ = 리 [1 2 1 a b a or M2i = [o 3 0 For example we have M₁+2i = (a) Show that M2₁ + M22 = Mz₁+z2 and cMz = Mez, for any complex numbers z, 21, 22 and real number c. (b) Show that Mz₁Mz2 = Mz₁22, for any complex numbers 21, 22. (c) Show that M₁/2 = M₂¹, 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).