True or False --- Justify the answer. a) The position of the leading 1\'s in a r

Question

True or False --- Justify the answer.
a) The position of the leading 1's in a row-echelon form of amatrix depend on how and in which order the rows have beenmultiplied during the application of the Gaussian algorithm. b) If a linear system has free variables (variables which donot correspond to a leading 1 in a row-echelon form of theaugmented matrix), the system has infinitely many solutions. True or False --- Justify the answer.
a) The position of the leading 1's in a row-echelon form of amatrix depend on how and in which order the rows have beenmultiplied during the application of the Gaussian algorithm. b) If a linear system has free variables (variables which donot correspond to a leading 1 in a row-echelon form of theaugmented matrix), the system has infinitely many solutions.

Explanation / Answer

Part a is False. It does not matterat all what order you do the operations in you should end up withthe same row-reduced matrix in all cases (assuming that you haveperformed the row operations correctly). You could give thesame matrix to a million people to row reduce and if they all dothe row operations correctly they will all end up with the sameending matrix so long as you reduce it completly. That's thebeauty of the Gauss-Jordan elimination method it will always giveyou the same answer. Part b is True. In order to solve a system of equationsfor an independent solution you need at least the same amountof linearly independent equations as variables. So if youhave 5 variables you will also need to have at least 5 linearlyindependent equations. This does not mean that you cannotsolve a system of equations with 5 variables with only 4equations. You can still solve this equations but you willnot get an independent solution. The "free" variable willcome to represent all real numbers and the other variables can beexpressed in terms of this "free" variable. This will giveyou an infinite number of combinations and thus an infinite numberof soultions. Example: 3v         -  y        = 0 8v               - 2z = 0         2x - 2y-   z = 0 So we have three equations but four variables. If we create a matrix to solve this equation and row reduce itwe will get this: [ 1 0 0 -1/4 0 ] [ 0 1 0    1/4 0 ] [ 0 0 1    3/4 0 ] So heres how we intrupret this data in terms of ourvariables. z = t = all real numbers y = (3/4)t x = (5/4)t v = (1/4)t So you can see that since t can be any number and all ourvariables depend on this we will have an infinite number ofsolutions. Hope this helps Good Luck

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