Trueor false ? Justify your answer ! (a)The positions of the leading 1’s in a ro

Question

Trueor false ? Justify your answer !

(a)The positions of the leading 1’s in a row-echelon form of a matrixdepend on how the rows have

beenmultiplied during the application of the Gaussianalgorithm.

(b)If a linear system has free variables (variables which do notcorrespond to a leading 1 in an rowechelon

formof the augmented matrix), the system has infinitely manysolutions.

(c)The general solution of a linear system is the explicit descriptionof all solutions of the system.

(d)We consider a linear system of 3 equations with 3variables. If the reduced row-echelon form ofthe

coefficient matrix of the system has aleading 1 ineach column, the system is solvable.

Explanation / Answer

ResponseDetails: Part A isFalse. It does not matter at all what order you do theoperations in you should end up with the same row-reduced matrix inall cases (assuming that you have performed the row operationscorrectly). You could give the same matrix to a millionpeople to row reduce and if they all do the row operationscorrectly they will all end up with the same ending matrix so longas you reduce it completly. That's the beauty of theGauss-Jordan elimination method it will always give you the sameanswer. Part B isTrue. In order to solve a system of equations for anindependent solution you need at least the same amount oflinearly independent equations as variables. So if you have 5variables 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. Part C isTrue. If you have a situation like the one we talkedabout in part B you will have this situation. Noticethat our solution for our example in part B gives just that anexplicit description of all solutions. Part D isTrue. Given that this discription is in fact decribingthe indentity matrix which tells us that all three equations arelinearly independent. This means that we have three variablesand three linearly independent equations meaning that it isindependtly solviable. Hope this helps Good Luck

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