## (Solution Download) If the row echelon form of A involves free variables

If the row echelon form of A involves free variables, then the system Ax = b will have infinitely many solutions.
In this case answer true if the statement is always true and false otherwise. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true. For example, consider the following statements about n × n matrices A and B.
(i) A + B = B + A
(ii) AB = BA
Statement (i) is always true. Explanation: The (i, j) entry of A + B is aij + bij and the (i, j) entry of B + A is bij + aij. Since aij + bij = bij + aij for each i and j, it follows that A + B = B + A.
The answer for statement (ii) is false. Although the statement may be true in some cases, it is not always true. To show this, we need only exhibit one instance where equality fails to hold. Thus, for example, if then This proves that statement (ii) is false.

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Question answered on Jul 22, 2018
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