1) Define logical equivalence.

2) What does the statement: The n x n matrix A is invertible.

3) How do I justify these statements: a) A is an invertible matrix b) A is row equivalent to the n x n identity matrix c) A has n pivot positions d) The equation Ax =0 has only the trivial solution. e) The equation Ax = b has at least one solution for each b in R

f) The columns of A span R^n g) The linear transformation x (arrow) Ax maps R^n onto R^n h) There is an n x n matrix D such that AD =I j) The columns of A form a basis of R^n

are equivalent to to the statement the n x n matrix A is invertable

2) What does the statement: The n x n matrix A is invertible.

3) How do I justify these statements: a) A is an invertible matrix b) A is row equivalent to the n x n identity matrix c) A has n pivot positions d) The equation Ax =0 has only the trivial solution. e) The equation Ax = b has at least one solution for each b in R

f) The columns of A span R^n g) The linear transformation x (arrow) Ax maps R^n onto R^n h) There is an n x n matrix D such that AD =I j) The columns of A form a basis of R^n

are equivalent to to the statement the n x n matrix A is invertable