Question 5: Download the dataset from https://drive.google.com/drive/folders/1A2B3C4D5E6F7G8H910J containing the files "algebraic_structures.xlsx" and "field_properties.txt." Instructions: 1. In "algebraic_structures.xlsx," each matrix is labeled with a structure name, such as group, ring, or field. 2. The file "field_properties.txt" lists properties of certain matrices and scalar values associated with each matrix. Answer the following: a) For any matrix labeled as a "field" in "algebraic structures.xlsx," determine whether it forms a field under standard matrix operations. Check for the presence of additive and multiplicative inverses and justify each step. b) For each matrix that forms a group under addition, verify if it is an abelian group. Describe the properties that characterize an abelian group and explain if the matrix G₁ from the dataset satisfies these properties. c) Using any matrix from the dataset that forms a ring, prove whether it has an identity element under multiplication. Discuss the significance of an identity element in a ring and the conditions under which a ring is a division ring. Show all steps in your derivation.

Explain the dimension of the row space.

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Find matrix X:   3.     6     5     4     -2   *X=     18     49     -20     6

Suppose that you are a pottery artist and want to create a special color to paint your work of art. Using the primary colors of red, green, and blue (or RGB) and the principles of linear algebra, combine the three primary colors to create new color schemes for painting your art work. Describe how you would mix your color mathematically. What subspace would you operate in? List your base vectors and describe them. If you want to vary the intensity of certain colors, how would you do it? What does vector addition do to your color scheme? What does positive scalar multiplication do to your color scheme? Give a vector representation of any color as an example. Explain the steps you undertake to make this color and name it.

Help me answer number 3 and 4