Indicate whether each statement is true or false a 5 points

Indicate whether each statement is true or false.

(a) (5 points True False If A is an n x n matrix with A -A, then A is not invertible

Solution

We knbow that det(AT) = det (A). Now if AT = -A, then det(AT) = det (A) = -det(A) so that det (A)= - det (A) or 2 det (A) = 0 or det (A) = 0. Then A is not invertible. The statement is True. The statement is False. If A is a nxn diagonalizable matrix, then A has n distinct linearly independent eigenvectors. There is no relation between diagionalizability and linear independence of rows of a matrix. The statement is True. We know that det(NT) = det (N). Now, if NT = N-1, then det(N) = det(NT) = det(N-1) = 1/det(N). Then det (NM2 ) = det(N)det(M2 ) = det(M2 )/det(N). The statement is True. T(v1+v2+v3) = T(v1) +T(v) +T(v3) = x+2x+3x = 6x. The statement is True. If A is a 5x5 diagonalizable matrix, then A has 5 distincrt linearly independent eigenvectors which will foirm a basis for R5. The statement is True. A.v = 0 when v is the zero vector which belongs to Rn.