Linear Algebra Answers

Prove these assertions concerning the effects of row operations on the determinant:

1) multiplying one row by k multiplies the determinant by k.
2) interchanging two rows changes the sign of the determinant.
3) adding to one row a multiple of another row has no effect on the determinant.

Let A be an N by N matrix with rank J<N. Prove then that there are N - J linearly independent solutions of the system Ax = 0, and the null space of A has dimension N - J.

Let A be an arbitrary M by N matrix and B the matrix in row-reduced echelon form obtained from A. Prove that there is a non-zero solution of the system of linear equations Ax = 0 iff B has fewer than N non-zero rows.

Prove that if A is M by N and M<N, then there is a non-zero x with Ax = 0.

Prove that any matrix A can be transformed into a matrix B in row-reduced echelon form using elementary row operations.

Research the Cayley-Hamilton Theorem and how it can be used to compute the inverse of a non-singular square matrix.

Why do we impose the requirement that the eigenvector be non-zero when we do not place this requirement on the eigenvalue.

Do row operations preserve the linear independence among the columns of a matrix? How about the rows of a matrix?

Prove that 0v = 0, for all v in V: I put down: from property 5 and property 8 we know that: v = 1v = (1+0)v = 1v + 0v = v + 0v = v+0 =v. thus, -v+v = -v+(v+0v) = (-v+v) + 0v. He said it was right but not totally right, and to prove it for anything (general case), ,and something about expressing W in terms of v in order to do that. Anyone?

Show that the rank of a matrix C = AB is never greater than the smaller of the rank of A and the rank of B. Can it ever be strictly less than the smaller of these two numbers?