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alina1380 [7]
3 years ago
15

Determine which matrices are in reduced echelon form and which others are only in echelon form. a. [Start 3 By 4 Matrix 1st Row

1st Column 1 2nd Column 0 3rd Column 0 4st Column 0 2nd Row 1st Column 0 2nd Column 2 3rd Column 0 4st Column 0 3rd Row 1st Column 0 2nd Column 0 3rd Column 1 4st Column 1 EndMatrix ]1 0 0 0 0 2 0 0 0 0 1 1 b. [Start 3 By 4 Matrix 1st Row 1st Column 1 2nd Column 0 3rd Column 1 4st Column 1 2nd Row 1st Column 0 2nd Column 1 3rd Column 1 4st Column 1 3rd Row 1st Column 0 2nd Column 0 3rd Column 0 4st Column 0 EndMatrix ]1 0 1 1 0 1 1 1 0 0 0 0 c. [Start 4 By 4 Matrix 1st Row 1st Column 0 2nd Column 0 3rd Column 0 4st Column 0 2nd Row 1st Column 1 2nd Column 3 3rd Column 0 4st Column 0 3rd Row 1st Column 0 2nd Column 0 3rd Column 1 4st Column 0 4st Row 1st Column 0 2nd Column 0 3rd Column 0 4st Column 1 EndMatrix ]

Mathematics
2 answers:
Y_Kistochka [10]3 years ago
7 0

Answer:

Step-by-step explanation:

Check the attachment for the solution

zhuklara [117]3 years ago
3 0

Answer:

  1. Echelon form.
  2. Reduced Echelon form.
  3. Neither.

Step-by-step explanation:

The objective is to determine which of the following matrices are in reduced echelon form and which others are only in echelon form. The given matrices are

                       \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0& 2 & 0 & 0 \\ 0& 0 & 1 & 1 \end{bmatrix},  \begin{bmatrix} 1 & 0 & 1 & 1 \\ 0& 1& 1 & 1 \\ 0& 0 & 0 & 0 \end{bmatrix}  and   \begin{bmatrix} 0& 0 & 0 & 0 \\ 1& 3 & 0 & 0 \\ 0& 0 & 1 & 0 \\ 0& 0 & 0 & 1 \end{bmatrix}.

First, recall what is an echelon and reduced echelon form of a matrix.

A matrix is said to be in a Echelon form if

  • If there is any zero rows, all nonzero rows are placed above them;
  • Each first non-zero entry in a row, which is the leading entry, is placed to the right of the leading entry of the row above it;
  • All elements below the leading entry must be equal to zero in each column.

A matrix is said to be in  a Reduced Echelon form if

  • In each non-zero row, the leading entry is 1.
  • In its column, each leading 1 is actually the only non-zero element.

A column that contains a leading 1 which is the only non-zero element is called a pivot column.

Now, let's have a look at the first matrix

                                 \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0& 2 & 0 & 0 \\ 0& 0 & 1 & 1 \end{bmatrix}

As we can see, it doesn't have any zero rows. Each leading entry in a row is placed to the right of the leading entry from the row above and all elements below the leading entries in all columns are equal to zero. Therefore, <u> this matrix is in an Echelon form.</u>

In the second row, the leading entry is 2, not 1, so because of the first property of the Reduced Echelon form, <u>it is not in a Reduced Echelon form. </u>

Notice that <em>it can be transformed to the Reduced Echelon form</em> by multiplying the second row by \frac{1}{2}.

The second matrix is

                                         \begin{bmatrix} 1 & 0 & 1 & 1 \\ 0& 1& 1 & 1 \\ 0& 0 & 0 & 0 \end{bmatrix}

There is a zero row, and all non-zero rows are placed above it. Each leading entry in a row, which is the first non-zero entry, is placed to the right of the entry of the row above it and all elements below the leading entry are equal to zero in each column, so <u>it is in the Echelon form</u>.

It is also <u>in the Reduced Echelon form</u>, since all non-zero rows the leading entry is 1 and it is the only non zero element in each column.

The least given matrix is

                                        \begin{bmatrix} 0& 0 & 0 & 0 \\ 1& 3 & 0 & 0 \\ 0& 0 & 1 & 0 \\ 0& 0 & 0 & 1 \end{bmatrix}

This matrix doesn't satisfy the condition that if there is any zero-row, it must be below all other non-zero rows, so <u>it is not in Echelon form.</u>

<em>A matrix that is not in an Echelon form, it is not in an Reduced Echelon form either. </em>

Therefore, <u>this matrix is not in an Reduced Echelon form.</u>

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