Question

In echelon form, which of the following is incorrect?
A. Every row of A which has all its entries O occurs below every row. Which has a non-zero entry.
B. The first non-zero entry in each non-zero row is!
C. The number of zeros before the first non-zero element is a row is less than the number of such zero in the new row.
D. Two rows have the same number of zeros before the first non-zero entry.

Answer 1

Hint: First we have to know about the properties of the echelon form of a matrix. In echelon form all non-zero rows are above any rows of all zeros. Each leading entry of a row is in a column to the right of the leading entry of the row above. Also, all entries of a column which is below a leading entry are zeros. So, we will use these properties to get the correct answer.

Complete step-by-step answer:
We have been asked to choose the incorrect statement about echelon form.
We know that the echelon has following properties:
1. All zero rows are at bottom.
2. The leading entry of each non-zero row after the first occurs to the right of the leading entry of the previous row.
3. The leading entry in any non-zero row is 1.
4. All entries in the column above and below leading 1 are zero.
Now let us take an example of an echelon form of a matrix as shown below:
\[A = \left( {\begin{array}{*{20}{c}}1&3&3\\0&1&2\\0&0&0\end{array}} \right)\]

By using the property and example of an echelon of a matrix, we get:
Option A is true.
Option B and option C are also true.
But, Option D is incorrect since the two rows have not the same numbers of zeros before the first non-zero.
Therefore, the correct option of the question is D.

Note: Just remember the characteristics of an echelon form of matrix so that you can easily solve these kinds of problems related to it.
When we come across such questions, it is very helpful to write down a matrix and try entering the elements in it according to each point. Then, compare if it matches with the properties of the echelon form of the matrix. Reading each option and understanding it becomes difficult, but writing it in matrix form makes things easier.