A matrix is in reduced row-echelon form if it meets four criteria:

1) No row comprising only 0's can be above a row with a nonzero entry.

2) The first nonzero entry in each nonzero row is a 1.

3) Each leading 1 is in a column to the right of the above leading 1.

4) In every column that includes a leading 1, all other entries are 0's.

The first nonzero entry in the second row is a 2, violating the second criterion:

is not in reduced row-echelon form.

Multiply both sides of the first equation by 2 in order to make the x-coefficients each other's opposite:

Add each side of this equation to each side of the other equation:

      , or

.

This indicates that the two equations are equivalent. Therefore,

The solution set can be written in parametric form as

, arbitrary.

For the product of two matrices to be defined, the number of columns in the first matrix must be equal to the number of rows in the second. This is the case, since has two columns and has two rows. is defined.

Matrices are multiplied by multiplying each row of the first matrix by each column of the second matrix. This is done by adding the products of the entries in corresponding positions. Thus,

,

the correct product.

A matrix is in reduced row-echelon form if it meets four criteria:

1) No row comprising only 0's can be above a row with a nonzero entry.

This condition is met, since the only all-zero row is the one at bottom:

2) The first nonzero entry in each nonzero row is a 1.

3) Each leading 1 is in a column to the right of the above leading 1.

Both conditions are met:

4) In every column that includes a leading 1, all other entries are 0's.

This condition is met:

meets all four criteria and is therefore in reduced row-echelon form.

A matrix is in reduced row-echelon form if it meets four criteria:

1) No row comprising only 0's can be above a row with a nonzero entry.

This is vacuously true, since there are no zero rows.

2) The first nonzero entry in each nonzero row is a 1.

3) Each leading 1 is in a column to the right of the above leading 1.

Both conditions are met:

4) In every column that includes a leading 1, all other entries are 0's.

Both conditions are met:

meets all four conditions and is therefore a matrix in reduced row-echelon form.

For the product of two matrices to be defined, the number of columns in the first matrix must be equal to the number of rows in the second. This is the case, since has two columns and has two rows. is defined.

Matrix multiplication is worked by multiplying each row of the first matrix by each column of the second matrix. This is done by adding the products of the entries in corresponding positions. Thus,

The product of any scalar value and a matrix is the matrix formed when each entry in the matrix is multiplied by that scalar. Thus, if

,

then

The statement is false, since the entry in Row 3, Column 1 is incorrect.

First, it must be established that is defined. This is the case if and only and have the same number of rows, which is true, and they have the same number of columns, which is also true. is therefore defined.

Addition of two matrices is performed by adding corresponding elements together, so

The statement is true.

First, it must be established that  is defined. This is the case if and only and have the same number of rows, which is true, and they have the same number of columns, which is also true.  is therefore defined.

Subtraction of two matrices is performed by subtracting corresponding elements together, so

The statement is true.

First, make the y-coefficients each other's opposite. This can be done by multiplying the first equation by 2 on both sides:

The y-coefficients of the two equations are now opposites, so, if the left and right sides of the two equations are added, the y-terms will cancel out, as follows:

The resulting statement is identically false. It follows that the two equations of the system are inconsistent with each other. The system has no solution.