State whether each statement is “True” or “False"

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State whether each statement is “True” or “False". Also, Justify each answer. If true, cite appropriate facts/theorems. If it is false, explain why/give a counterexample that shows why the statement is not necessarily true in all cases.

a. Every matrix is row equivalent to a unique matrix in echelon form.

b. Any system of n linear equations in n variables has at most n solutions.

c. If a system of linear equations has two different solu- tions, it must have infinitely many solutions.

d. If a system of linear equations has no free variables, then it has a unique solution.

e. If an augmented matrix A b ] is transformed into [C d ] by elementary row operations, then the equa- tions Ax = b and Cx = d have exactly the same solu- tion sets.

f. If a system Ax = b has more than one solution, then so does the system Ax = 0.

g. If A is an mxn matrix and the equation Ax = b is consistent for some b, then the columns of A span R".

h. If an augmented matrix [A b] can be transformed by elementary row operations into reduced echelon form, then the equation Ax=b is consistent.

i. If matrices A and B are row equivalent, they have the same reduced echelon form.

j. The equation Ax = 0 has the trivial solution if and only if there are no free variables. Home page

k. If A is an mxn matrix and the equation Ax = b is con- sistent for every b in R", then A has m pivot columns.

l. If an mxn matrix A has a pivot position in every row, then the equation Ax=b has a unique solution for each b in R".

m. If an n xn matrix A has n pivot positions, then the reduced echelon form of A is the n x n identity matrix.

n. If 3 x 3 matrices A and B each have three pivot posi- tions, then A can be transformed into B by elementary row operations.