, the inverse of a two-by-two matrix

 ,

can be calculated as follows:

,

where .

Setting each of the values accordingly,

, or

.

For the sum of two matrices to be defined, they must have the same number of rows and columns. is a matrix with three columns; since, in this problem, refers to the two-by-two identity matrix

,

has two columns. Since the number of columns differs, is undefined.

The product of two matrices and , where has rows and columns and has rows and columns, is a matrix with rows and columns. It follows that must have the same number of rows as . Since has three rows, so does . Nothing can be inferred about the number of rows of .

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

This is not equal to the matrix

,

since the entries in the second and third rows differ. The statement is false.

For two matrices to be equal, two conditions must hold:

1) The matrices must have the same dimension. This can be seen to be the case, since both matrices have two rows and one column.

2) All corresponding entries must be equal. For this to happen, it must hold that

This is a system of two equations in two variables, which can be solved as follows:

Add both sides of the equations:

It follows that

Substitute back:

Thus, there exists a solution to the system, and, consequently, to the original matrix equation. The statement is therefore false.

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.