Instruction: Form a 3x3 matrix A based on the first nine digits of your student ID. For example, if your student ID is 1234567890 then form the matrix 2 A = 5 8 et C1, C2, and cy denote the column vectors of A and aj the entry of A in row i and column j. For example, with A as above, a = ( # ) . " - (! ) . - -(: ) Further denote the sum of the first six digits of your student ID by k and the sum of the first three digits by d. In case both numbers are equal to each other, increase & by 2. In the above example, k =1+2+3+4+5+6=21 and d=1+2+3=6. Given your set-up for A, dij, C1, C2, Cs, k and d compute the following. Show your work so an interested peer can follow along. Part 1: Vectors a. Find c1 + 413 . C2. b. Find C1 . C2. C. Find the angle between c, and C2. d. Describe the plane orthogonal to the vector c3. Part 2: Matrices a. Find the determinant of A b. Find the trace of A. C. Find B = A . AT. Is B symmetric? d. Find the inverse of A. Part 3: Basic combinatorics An ice cream shop has k different flavors of ice cream. How many different ways are there to select d scoops of ice cream in each of the following scenarios? k and d are the numbers you calculated above. Make sure your exposition demonstrates an understanding of the argument and formulas used, not just applying the formula to each instance a. No flavor can be repeated and the order in which the scoops are chosen matter. E.g., you cannot have three scoops, including vanilla, vanilla, and strawberry. Also, having chocolate at the bottom and strawberry on top is different from having strawberry at the bottom and chocolate on top. b. No flavor can be repeated and the order in which the scoops are chosen does not matter. E.g., you cannot have three scoops, including vanilla, vanilla, and strawberry. Also, having chocolate at the bottom and strawberry on top is the same as having strawberry at the bottom and chocolate on top. C. Flavors can be repeated and the order in which the scoops are chosen matters. E.g., you can have three scoops, including vanilla, vanilla, and strawberry. Also, having chocolate at the bottom and strawberry on top is different from having strawberry at the bottom and chocolate on top. d. Flavors can be repeated and the order in which the scoops are chosen does not matter. E.g., you can have three scoops, including vanilla, vanilla, and strawberry. Also, having chocolate at the bottom and strawberry on top is the same as having strawberry at the bottom and chocolate on top. Part 4: Systems of linear equations. a. Write out explicitly the system of equations implied by the matrix equation