An elementary matrix is one that can be formed by performing a single row operation on the (here, four-by-four) identity matrix

A matrix qualifies as an elementary matrix if and only if it differs from  in exactly one of three ways:

1) Exactly two rows have changed position with each other.

2) Exactly one of its diagonal elements - the element in Row 1 and Column 1, Row 2 and Column 2, and so forth - is equal to a nonzero number other than 1.

3) Exactly one of its nondiagonal elements is equal to a nonzero number.

 violates these criteria because all four of its diagonal elements are equal to nonzero numbers other than 1.

An elementary matrix is one that can be formed by performing a single row operation on the (here, four-by-four) identity matrix

A matrix qualifies as an elementary matrix if and only if it differs from  in exactly one of three ways:

1) Exactly two rows have changed position with each other.

2) Exactly one of its diagonal elements - the element in Row 1 and Column 1, Row 2 and Column 2, and so forth - is equal to a nonzero number other than 1.

3) Exactly one of its nondiagonal elements is equal to a nonzero number.

 satisfies these criteria in that it differs from  only in that one diagonal element (Row 1, Column 1) has been changed to the nonzero number 0.000001.

An elementary matrix is one that can be formed by performing a single row operation on the (here, four-by-four) identity matrix

A matrix qualifies as an elementary matrix if and only if it differs from  in exactly one of three ways:

1) Exactly two rows have changed position with each other.

2) Exactly one of its diagonal elements - the element in Row 1 and Column 1, Row 2 and Column 2, and so forth - is equal to a nonzero number other than 1.

3) Exactly one of its nondiagonal elements is equal to a nonzero number.

 violates these criteria because three of its nondiagonal elements - the first three elements in its fourth row - are equal to nonzero numbers. 

 is an elementary matrix in that it can be formed from the (five-by-five) identity matrix  by one row operation.  differs from  in that  appears instead of 0 in Row 4, Column 1; therefore, the row operation that has been performed on  to obtain  is

Premultiplying  by  has the effect of performing the same row operation on , so, to each element in row 4 of , add  times the corresponding element of , while leaving all other elements intact:

 is an elementary matrix in that it can be formed from the (five-by-five) identity matrix  by one row operation.  differs from  in that Row 1 and Row 5 have been switched; therefore, the row operation that has been performed on  to obtain  is .

Premultiplying  by  has the effect of performing the same row operation on , so, in , switch Row 1 and Row 5, while leaving all other elements intact:

,

the correct choice.

A matrix is considered to be in reduced row-echelon form if and only if it meets all of the following criteria:

1) Any rows comprising only "0" elements must be gathered at the bottom of the matrix.

2) The first nonzero element in each other row must be a "1". 

3) The leading "1" in each row must be to the right of the leading "1" of the above row.

The matrix 

satisfies the first two criteria. However, the third condition is violated, as the leading "1" in the third row is directly under that in the row above it. Therefore, this matrix is not in reduced row-echelon form. 

A matrix is considered to be in reduced row-echelon form if and only if it meets all of the following criteria:

1) Any rows comprising only "0" elements must be gathered at the bottom of the matrix.

2) The first nonzero element in each other row must be a "1". 

3) The leading "1" in each row must be to the right of the leading "1" of the above row.

The matrix 

can be seen to satisfy all three conditions and it is therefore in reduced row-echelon form.

A matrix is considered to be in reduced row-echelon form if and only if it meets all of the following criteria:

1) Any rows comprising only "0" elements must be gathered at the bottom of the matrix.

2) The first nonzero element in each other row must be a "1". 

3) The leading "1" in each row must be to the right of the leading "1" of the above row.

The matrix 

An elementary matrix is one that can be formed by performing a single row operation on the (here, four-by-four) identity matrix

A matrix qualifies as an elementary matrix if and only if it differs from  in exactly one of three ways:

1) Exactly two rows have changed position with each other.

2) Exactly one of its elements on its main diagonal - upper left to lower right - is equal to a nonzero number other than 1.

3) Exactly one of its nondiagonal elements is equal to a nonzero number.

 satisfies this condition in that it differs from  in only one of these ways - one non-diagonal element (Row 4, Column 1) has been changed to a nonzero number. This is an elementary matrix.

A matrix is considered to be in reduced row-echelon form if and only if it meets all of the following criteria:

1) Any rows comprising only "0" elements must be gathered at the bottom of the matrix.

2) The first nonzero element in each other row must be a "1". 

3) The leading "1" in each row must be to the right of the leading "1" of the above row.

The matrix 

violates the first condition, as the only zero row in this matrix is above a nonzero row. This matrix is not in reduced row-echelon form.