Basis maths-linear algebra-Q10

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In a system of linear equations, in the Augmented matrix, a leading entry of a row refers to the left most nonzero entry (if any).

Consider the following statements :

1. The Leading entries in any row are always in the same positions in any Echelon form obtained from a given matrix.

2. Elementary row operations on an augmented matrix never change the solution set of the associated linear system.

Which of the above statements is/are True?

(A). 1 Only

(B). 2 Only

(C). Both

(D). None

asked May 15 in Basic Maths by gbeditor (44,560 points)
reshown May 16 by gbeditor

1 Answer

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Statement 1 : When row operations on a matrix produce an echelon form, further row operations to obtain the reduced echelon form do not change the positions of the leading entries. Since the reduced echelon form is unique,the leading entries are always in the same positions in any echelon form obtained from a given matrix. These leading entries correspond to leading 1's in the reduced echelon form.

Statement 2: Suppose a system is changed to a new one via row operations. By considering each type of row operation, you can see that any solution of the original system remains a solution of the new system. Conversely, since the original system can be produced via row operations on the new system, each solution of the new system is also a solution of the original system.
answered May 16 by deepak-gatebook (112,760 points)
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