For each of the following statements, determine if the statement is Always True, Sometimes True, or...

  

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For each of the following statements, determine if the statement is Always True, Sometimes True, or Always False. No justification is required for a correct answer. However, partial credit for an incorrect answer is only possible if reasonable justification is provided. (a) (2 points) If A is an m x n matrix and b in R cannot be written as a linear combination of the columns of A, then the matrix equation Ax = b has at least one solution. (b) (1 points) Let A be a 5 x 5 matrix, which has eigenvalues A=-2, A-1, A=0, A = 1, and A 2. Then A is diagonalizable. (c) (2 points) Let M be a 7 x 11 matrix. Then rank(M) = 7. (d) (1 points) Suppose that M is a 3 x 3 matrix, which can be factored as a product M = [-11 0 01 PDP-1, where D = 020 . Then det(M) = 66. 0 0-3 (e) (2 points) Let u and v be vectors in R" and suppose u-v = 1. Then u and v are orthogonal vectors. For each of the following statements, determine if the statement is Always True, Sometimes True, or Always False. No justification is required for a correct answer. However, partial credit for an incorrect answer is only possible if reasonable justification is provided. (a) (2 points) If A is an m x n matrix and b in R cannot be written as a linear combination of the columns of A, then the matrix equation Ax = b has at least one solution. (b) (1 points) Let A be a 5 x 5 matrix, which has eigenvalues A=-2, A-1, A=0, A = 1, and A 2. Then A is diagonalizable. (c) (2 points) Let M be a 7 x 11 matrix. Then rank(M) = 7. (d) (1 points) Suppose that M is a 3 x 3 matrix, which can be factored as a product M = [-11 0 01 PDP-1, where D = 020 . Then det(M) = 66. 0 0-3 (e) (2 points) Let u and v be vectors in R" and suppose u-v = 1. Then u and v are orthogonal vectors.