QUESTION 1

Let A be a 6 x 6 matrix with det(A) = 35 and B be the matrix B = -3AA". Definition: A matrix C is said to be skew-symmetric if CT = -C. (a) Select ALL options that are TRUE. The matrix B is symmetric The matrix B is skew-symmetric The matrix B is singular The matrix B is invertible (b) What is the value of det(A]) ? det (A' ) = Number (c) What is the value of det(BA-]) ? det(BA )) = NumberJeremy enjoys using Gaussian elimination to evaluate the determinant of a matrix and the latest challenge is the following matrix. k k k - 3 Let A = 2k +7 k - 3 where k E R . k k - 3 Jeremy performs the following operations using Maple: > with (LinearAlgebra) : > A := | | >: > B : = ColumnOperation (A, 3, 1/ (k-3) ) ; k B:= 2k + 7 k > C := ColumnOperation (B, [2,3] , -k) ; k 0 C := k +7 6 0 (a) Find a such that |A| = a| B). a = (b) Find S such that | B| = BICI. B = (c) Determine the value(s) of k so there exists a non-zero solution x to Ax = 0. Syntax advice: Write your answer as a list or set of real constant(s) in order, separated by commas. For example, enter 1, 2, 3, 4 or {1, 2, 3, 4}. Do NOT put a comma at the end of your answer and do NOT write two commas without a number between them. Enter exact numbers. Do not use decimal approximations. Please use the "Preview" button to check your syntax. KE (d) For your largest value of & determine a non-zero solution x to Ax = 0. Syntax advice: Enter your answer as vector. To enter the vector , use the Maple syntax . Please use the "Preview" button to check your syntax. X = (e) For your smallest value of & determine a non-zero solution x to Ax = 0. Syntax advice: Enter your answer as vector. To enter the vector CON - , use the Maple syntax . Please use the "Preview" button to check your syntax. X =