5 Problem 5-Showing that complex numbers, C, have a matrix form and have a visual representation that looks like R² For complex number in standard form a + bi, a is called the real component (the one without an i) and b is called the imaginary component (the one with an i) This question will show that a+bi can be expressed as the matrix (" a and that complex number operations correspond directly to matrix addition and multiplication Consider the set of all 2 x2 matrices with real entries of the following form a b (ε -). where a and b are any real numbers 5.1 a) Show that the set is closed under matrix addition, matrix multiplication and scalar multiplication. (General) 5.2 b) CL 5.2 b) Show that matrix multiplication is commutative in this set (General) 5.3 c) Show explicitly (without the determinant) that any product is the zero matrix only if one of the matrices are all zero i.e solve and show that one matrix must be all zeroes ( - ) ( 2 ) - (88) (You should have at the end a²=-62 or a² = y² which of course implies 5.4 d) Show that all matrices have an inverse if a, b are not both equal to zero (Gen- eral) 5.5 e) For all matrices of this form with determinant equal to 1, describe a and b as functions of a single variable 0. (Hint: we have talked about this matrix before) 5.6 f) Take the product of the complex numbers 2+ 3i and 3-5i. Express in a+bi form (recall i² = -1) are the real and imaginary components 5.7 g) 2³) (3³ 2). Then take the product (3 Show that the first column() of the product a + bi in f) 5.8 h) Then show this in general: (a+bi)(x+yi) and (a+bi)+(x+yi) 5.8 h) Then show this in general: (a+bi)(x+yi) and (a + bi) + (x+yi) corresponds directly to a and a ( ) ( ) ( )+(7) (i) C a 5.9 i) Find the Eigenvalues of (%) -") a (Keep it general. Hint -62 = bi and the Eigenvalues will be complex if