Question

True or False, provide a brief reason or provide a counterexample for the "False"

1) If an n by n matrix is diagonalizable, then it must have n different eigenvalues. 2) If A is a square matrix and det(A) = 0, then A must have a row of 0s. 3) If A is positive definite, then A^-1 is positive definite. 4) Asymmetric matrix with a positive determinant is positive definite. 5) The following matrix is positive definite: a = [1 2 0 0 2 6 -2 0 0 -2 5 -2 0 0 -2 3].

Show transcribed image text 1) If an n by n matrix is diagonalizable, then it must have n different eigenvalues. 2) If A is a square matrix and det(A) = 0, then A must have a row of 0s. 3) If A is positive definite, then A^-1 is positive definite. 4) Asymmetric matrix with a positive determinant is positive definite. 5) The following matrix is positive definite: a = [1 2 0 0 2 6 -2 0 0 -2 5 -2 0 0 -2 3].