Thank you for the opportunity to help you with your question!

#10 is true.

#11. is False. The rank of a matrix is simply not the number of its nonzero rows. The rank of a matrix is the number of leading 1's in its reduced row echelon form.

#12.FALSE. Even though matrix A is transformed by elementary row operations it may NOT be in reduced row echelon form. The rank must be the number of nonzero rows in reduced row echelon form.

#13. TRUE. We have the matrix in reduced row echelon form, therefore the number of nonzero rows is the rank of matrix A. Even though nonzero, we can divide that row by that number to get the leading 1.

#14.TRUE.

#15.TRUE

#16. TRUE

#17. TRUE.

#18.TRUE.

#19.FALSE

#20.TRUE

Please let me know if you need any clarification. I'm always happy to answer your questions.

Nov 12th, 2015

Sorry, I need to recorrect #20. It should be FALSE, because in order for the columns to be linearly independent, the zero solution is the only solution to equation Ax = 0, the TRIVIAL solution is the only solution to that system in order for it to be linearly independent.