#5 Mark each of the following statements as either True or False, and justify yo
ID: 1719564 • Letter: #
Question
#5 Mark each of the following statements as either True or False, and justify your answer. (If true, eve the locution in the textbook where the statement appears, or refer to a definition or theorem. If false, give a specific example that shows the statement is not true in all cases.) Elementary row operations on an aumented matrix never change the solution set of the associated linear system. Two matrices are row equivalent if they have the same number of rows. Two linear systems are equivalent if they have the same solution set. Whenever a system has free variables. the solution set contains many solutions. What would you have to know about the pivot columns in an augmented matrix in order to know that the linear system is consistent and has a unique solution?Explanation / Answer
A condition should be that once this diagonalized matrix, the amount nonzero elements on the main diagonal has to be the amount of system variables.
For a given number of unknowns, the number of solutions to a system of linear equations depends only on the rank of the matrix representing the system and the rank of the corresponding augmented matrix. Specifically, according to the Rouché–Capelli theorem, any system of linear equations is inconsistent (has no solutions) if the rank of the augmented matrix is greater than the rank of the coefficient matrix; if, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has k free parameters where k is the difference between the number of variables and the rank; hence in such a case there are an infinitude of solutions.
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