Determine whether the following statements are true or false, in general, and justify your answers. (1) For any (square or not) matrix A, a system Ax = 0 of linear equations has at least one solution. (2) A system Ax = b of linear equations has a solution if and only if b = 0. (3) Let A and B be row-equivalent square matrices. Then A is invertible if and only if B is invertible. (4) For any two (square or not) matrices A and B, if AB is the identity, then BA is also the identity. (5) Every permutation matrix is invertible. (6) Let A be a square matrix such that AA = A. Then A is the identity. (7) If A and B are invertible matrices such that A? I and B2 I, then (AB) -1 = BA. (8) If A and B are invertible matrices, A + B is also invertible. b 07 a b 07 (9) 0 bis symmetric and 0-6 is skew-symmetric. 0 a 0 (10) If A, B and AB are all symmetric, then AB = BA. (11) The difference A - B of any two symmetric matrices A and B is also sym- metric. (12) The product of any two skew-symmetric matrices is either symmetric or skew- symmetric. (13) If A is invertible and symmetric, then A - is also symmetric. (14) Let ABT = I. Then A is invertible if and only if B is invertible. square matrix A is not invertible, then neither is AB for any B. (16) If E, and E, are elementary matrices, then E1 E2 = E.E. (17) The product of any two elementary matrices is again an elementary matrix. (18) The product of any number of elementary matrices is invertible. ) (19) The inverse of an invertible upper triangular matrix is also upper triangular. a -C -a - (15) If a