Is echelon form a requirement to show a matrix has no solutions?












2














Do I need to get a matrix into echelon form to prove that it has no solutions, or do I only need a pivot in the last column. For example, would the following row operation show that there are no solutions to the linear system represented that by the augmented matrix below?



$begin{bmatrix}1&-3&0&5\-1&1&5&2\-1&1&5&3end{bmatrix}$ $Rightarrow$ $begin{bmatrix}1&-3&0&5\-1&1&5&2\0&0&0&1end{bmatrix}$






linear-algebra matrices







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  • What entry did you pivot on?
    – coffeemath
    7 hours ago










  • I'm not sure what definition of pivot you are using. By pivot, I mean the "leading entry in a row".
    – Gaussian Elimination
    7 hours ago
















2














Do I need to get a matrix into echelon form to prove that it has no solutions, or do I only need a pivot in the last column. For example, would the following row operation show that there are no solutions to the linear system represented that by the augmented matrix below?



$begin{bmatrix}1&-3&0&5\-1&1&5&2\-1&1&5&3end{bmatrix}$ $Rightarrow$ $begin{bmatrix}1&-3&0&5\-1&1&5&2\0&0&0&1end{bmatrix}$






linear-algebra matrices







share|cite|improve this question







New contributor




Gaussian Elimination is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




















  • What entry did you pivot on?
    – coffeemath
    7 hours ago










  • I'm not sure what definition of pivot you are using. By pivot, I mean the "leading entry in a row".
    – Gaussian Elimination
    7 hours ago














2












2








2


1





Do I need to get a matrix into echelon form to prove that it has no solutions, or do I only need a pivot in the last column. For example, would the following row operation show that there are no solutions to the linear system represented that by the augmented matrix below?



$begin{bmatrix}1&-3&0&5\-1&1&5&2\-1&1&5&3end{bmatrix}$ $Rightarrow$ $begin{bmatrix}1&-3&0&5\-1&1&5&2\0&0&0&1end{bmatrix}$






linear-algebra matrices







share|cite|improve this question







New contributor




Gaussian Elimination is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











Do I need to get a matrix into echelon form to prove that it has no solutions, or do I only need a pivot in the last column. For example, would the following row operation show that there are no solutions to the linear system represented that by the augmented matrix below?



$begin{bmatrix}1&-3&0&5\-1&1&5&2\-1&1&5&3end{bmatrix}$ $Rightarrow$ $begin{bmatrix}1&-3&0&5\-1&1&5&2\0&0&0&1end{bmatrix}$






linear-algebra matrices




linear-algebra matrices






share|cite|improve this question







New contributor




Gaussian Elimination is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question







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Gaussian Elimination is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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share|cite|improve this question




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asked 7 hours ago









Gaussian Elimination

112




112




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  • What entry did you pivot on?
    – coffeemath
    7 hours ago










  • I'm not sure what definition of pivot you are using. By pivot, I mean the "leading entry in a row".
    – Gaussian Elimination
    7 hours ago


















  • What entry did you pivot on?
    – coffeemath
    7 hours ago










  • I'm not sure what definition of pivot you are using. By pivot, I mean the "leading entry in a row".
    – Gaussian Elimination
    7 hours ago
















What entry did you pivot on?
– coffeemath
7 hours ago




What entry did you pivot on?
– coffeemath
7 hours ago












I'm not sure what definition of pivot you are using. By pivot, I mean the "leading entry in a row".
– Gaussian Elimination
7 hours ago




I'm not sure what definition of pivot you are using. By pivot, I mean the "leading entry in a row".
– Gaussian Elimination
7 hours ago










3 Answers
3






active

oldest

votes


















3














Your transformation suffices; if you translate your matrix back into a system of linear equations, the last row gives the equation $0 cdot x + 0 cdot y + 0 cdot z = 1$.



This is indeed the beauty of the representation of systems of linear equations as matrices: you perform simple operations on the rows of the matrix but preserve the solution set of the original system. At the end, you just read off the solutions.






share|cite|improve this answer





























    2














    No, it's not required. Once you've performed row operations (as you have) to a point where you reach an inconsistency, you can conclude the system has no solutions. Going all the way to echelon form, or rref, won't change that.






    share|cite|improve this answer





























      2














      You only have to prove the matrix of the homogeneous part and the augmented matrix do not have the same rank. In the present case, it is enough: the matrix of the homogeneous part has rank $2$ and the augmented matrix has rank $3$.






      share|cite|improve this answer





















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        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        3














        Your transformation suffices; if you translate your matrix back into a system of linear equations, the last row gives the equation $0 cdot x + 0 cdot y + 0 cdot z = 1$.



        This is indeed the beauty of the representation of systems of linear equations as matrices: you perform simple operations on the rows of the matrix but preserve the solution set of the original system. At the end, you just read off the solutions.






        share|cite|improve this answer


























          3














          Your transformation suffices; if you translate your matrix back into a system of linear equations, the last row gives the equation $0 cdot x + 0 cdot y + 0 cdot z = 1$.



          This is indeed the beauty of the representation of systems of linear equations as matrices: you perform simple operations on the rows of the matrix but preserve the solution set of the original system. At the end, you just read off the solutions.






          share|cite|improve this answer
























            3












            3








            3






            Your transformation suffices; if you translate your matrix back into a system of linear equations, the last row gives the equation $0 cdot x + 0 cdot y + 0 cdot z = 1$.



            This is indeed the beauty of the representation of systems of linear equations as matrices: you perform simple operations on the rows of the matrix but preserve the solution set of the original system. At the end, you just read off the solutions.






            share|cite|improve this answer












            Your transformation suffices; if you translate your matrix back into a system of linear equations, the last row gives the equation $0 cdot x + 0 cdot y + 0 cdot z = 1$.



            This is indeed the beauty of the representation of systems of linear equations as matrices: you perform simple operations on the rows of the matrix but preserve the solution set of the original system. At the end, you just read off the solutions.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 7 hours ago









            MacRance

            956




            956























                2














                No, it's not required. Once you've performed row operations (as you have) to a point where you reach an inconsistency, you can conclude the system has no solutions. Going all the way to echelon form, or rref, won't change that.






                share|cite|improve this answer


























                  2














                  No, it's not required. Once you've performed row operations (as you have) to a point where you reach an inconsistency, you can conclude the system has no solutions. Going all the way to echelon form, or rref, won't change that.






                  share|cite|improve this answer
























                    2












                    2








                    2






                    No, it's not required. Once you've performed row operations (as you have) to a point where you reach an inconsistency, you can conclude the system has no solutions. Going all the way to echelon form, or rref, won't change that.






                    share|cite|improve this answer












                    No, it's not required. Once you've performed row operations (as you have) to a point where you reach an inconsistency, you can conclude the system has no solutions. Going all the way to echelon form, or rref, won't change that.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 7 hours ago









                    pwerth

                    1,609411




                    1,609411























                        2














                        You only have to prove the matrix of the homogeneous part and the augmented matrix do not have the same rank. In the present case, it is enough: the matrix of the homogeneous part has rank $2$ and the augmented matrix has rank $3$.






                        share|cite|improve this answer


























                          2














                          You only have to prove the matrix of the homogeneous part and the augmented matrix do not have the same rank. In the present case, it is enough: the matrix of the homogeneous part has rank $2$ and the augmented matrix has rank $3$.






                          share|cite|improve this answer
























                            2












                            2








                            2






                            You only have to prove the matrix of the homogeneous part and the augmented matrix do not have the same rank. In the present case, it is enough: the matrix of the homogeneous part has rank $2$ and the augmented matrix has rank $3$.






                            share|cite|improve this answer












                            You only have to prove the matrix of the homogeneous part and the augmented matrix do not have the same rank. In the present case, it is enough: the matrix of the homogeneous part has rank $2$ and the augmented matrix has rank $3$.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered 7 hours ago









                            Bernard

                            118k638112




                            118k638112






















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