7. Let A, B and C be n x n matrices and let D = ABC. Prove that if D is nonsingular, then A, B and C must be nonsingular.

Let A,D, and P be nn matrices satisfying AP=PD. Assume that P is nonsingular and solve this for A. Must it be true that A=D?

Let A and B be square matrices of order n satisfying, Ax=Bx for all x in all Rn. a Find the rank and nullity of AB. b Show that matrices A and B must be identical.

Let A and B be square matrices of order 4 such that |A|=5 and |B|=3.Find a |A2|, b |B2|, c |A3|, and d |B4|

Consider an mn matrix A and an np matrix B. Show that the row vectors of AB are in the row space of B and the column vectors of AB are in the column space of A.

Determine if the statement is true or false. If the statement is false, then correct it and make it true. For the product of two matrices to be defined, the number of rows of the first matrix must equal the number of columns of the second matrix.

Let A and B be square matrices of order n over Prove or disprove that the product AB is a diagonal matrix of order n over if B is a diagonal matrix.

We can add (or subtract) two matrices only if they have the same ______.