Upload to Gradescope 1. [Basic Matrix Operations] If an operation cannot be done for some reason (e.g. the dimensions of two matrices do not allow for them to be multiplied together), write "This operation cannot be done because." (a) Consider the matrices: e jn/2 e j3x/2 3e ja eJ#/2 For any complex numbers, put the answer in rectangular form. Note that it might be worthwhile to use your (vast) knowledge of complex numbers to re-write some of the elements of A in simpler form before performing the following operations. i. Find A - B. ii. Find A + jB. iii. Find AB. iv. Find ATB". v. Find B' A"by doing matrix multiplication directly (do not use the property of the transpose of a product of two matrices here; that is the next part). vi. Which answer above (iv or v) is equal to the transpose of your answer from ini; that is, which is equal to (AB)". Is this what you expected based on the property from class? (b) Now, still with the same matrix A as above, consider: c = - 2 d = [5 -1 -4] i. Find Ac. ii. Find cA iii. Find cfA. iv. Find c A . (You can use a property here if you want.) v. Find Ad. vi. Find Ad". vii. Find ed. viii. Find dc.2. [Gaussian Elimination] It is important to remember that the properties of matrices that we are (and will be) studying are important largely for what they tell us about the solution of linear equations. It is easy to get caught up in all of the manipulations of matrices and forget what they mean and why we care. This problem will help with that. In high school, you solved systems of linear equations often by substitution: (1) solve one equation for one of the variables; (2) substitute the result of Step 1 into the other equation(s); (3) repeat until you have a single equation with a single variable; (4) solve for that single variable; (5) plug the result into your equations to solve for the other variables. (a) Solve the following systems of two equations and two unknowns by substitution: i. 201 + 12 = 3 1 + 2 = 1 1i. 1 3.r1 + 3.2 = 3 iii. 1 3x1 + 3.2 = 4 (b) Repeat part (a) using Gaussian elimination on the augmented matrix A, as defined in class. (c) For each of the systems in part (a), find the rank of the coefficient matrix A, as defined in class. Indicate what your answer tells you about the number of solutions for each of the systems. (d) Solve the following systems of three equations and three unknowns by employing Gaussian elimination. i. 2x1 + 212 + 313 7 I1 + 212 + 13 = 2 0 ii. 6 2x1 + 402 + 203 = 12 mtxt = 2 iii. $1 + 202 + 3x3 6 2x1 + 402 + 203 = 16 2x1 + 402 = 12 (e) For each of the systems in part (d), find the rank of the coefficient matrix A, as defined in class. Indicate what your answer tells you about the number of solutions for each of the systems