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PROBLEM 1 PROBLEM 4 Choose one of the following two options to complete: Solve the system Ax = b where Option 1 Perform Gaussian elimination on the matrix and and b= by solving gLc=bandUx=c -112 0 1 PROBLEM to transform the matrix into an upper triangular matrix. Show Give an example of a nonzero matrix A such that A2 = 0. Is the elimination through multiplications of elimination and it possible for AT A = 0 for any nonzero matrix. Explain your permutation matrices with the matrix A. reasoning. Option 2 PROBLEM 6 Given a square, non-singular matrix (array) A of real numbers, Construct an argument that proves (AB)T = BTAT by use a language such as Python, MATLAB, etc to write your own function that performs a simplified version of Gaussian looking at arbitrary entries aij, big, etc of the matrices and their transposes. elimination to reduce the matrix to an upper triangular matrix U. The program will not need to perform any permutations. Let the function have the following format: PROBLEM 7 Prove that A is invertible if a # 0 and a # b (find the pivots U = gauss(A) or A-1). Then find three numbers C so that C is not invertible. PROBLEM 2 If S1 and S2 are symmetric matrices, when will S1 + S2 be a symmetric matrix? Write the condition(s) and show that the A= ... a and sum is symmetric. COMMENTS PROBLEM 3 You are encouraged to use technology to verify your work, If possible, find the LU and LDU factorizations of the matrix but the nature of this class requires you to demonstrate your 101 written calculations. Give credit to sources that you use other A 222 than our course textbook and notes. Please, report any typos 345 or mistakes.

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