1.4 Gaussian Elimination T/F

If the reduced row echelon form of [A b] contains a zero row, then Ax = b must have infinitely many solutions.

False it is possible if the system contains more equations as variables that the system contains one solution. It is also possible that the system has no solutions if the reduced row echelon form also contains a row with all zeros except for the last column.

If the reduced row echelon form of [A b] contains a zero row, then Ax = b must be consistent.

False, a system is consistent if the reduced row echelon form of the augmented matrix contains a row with all zeros except for the last column. The reduced row echelon form containing a zero row is thus not a sufficient condition.

If a system of m linear equations in n variables is equivalent to a system of p linear equations in q variables, then m = p.

False, for example if the first system of linear equations contains twice the same row and the second system of equation only contains this equation once.

There is a unique sequence of elementary row operations that transforms a matrix into its reduced row echelon form.

False, for example multiplying a row by two is an elementary row operation, but you can obtain the same by adding the row to itself.

The sum of the rank and nullity of a matrix equals the number of rows in the matrix.

False, the sum of the rank and nullity of a matrix equals the number of columns in the matrix.

The third pivot position in a matrix lies in column 3.

False, the third pivot position of a matrix always lies in the third row, but in which column it lies is not in advance determined.

There exists a 5 x 8 matrix with rank 3 and nullity 2.

False; the nullity is the number of columns 8 decreased by the rank 3, this it is 8 - 3 = 5.

If the equation Ax = b is inconsistent, then the rank of [A b] is greater than the rank of A.

True, The augmented matrix [A b] is consistent if the matrix contains a row with all zeros except for in the last columns and the corresponding matrix A then has in this row all zeros. Thus we then know that the rank of the augmented matrix is one greater than the rank of A.

A column of a matrix A is a pivot column if the corresponding column in the reduced row echelon form of A contains the leading entry of some nonzero row.

True, a pivot column is a column that contains a leading entry in the reduced row echelon form.

If some column of matrix A is a pivot column, then the corresponding column in the reduced row echelon form of A is a standard vector.

True, because a pivot column contains a 1 on the pivot position and the rest of the column contains zeros, thus the pivot column is the column of an identity matrix. By definition of standard vectors, we then know that the pivot column has to be a standard vector.

If a system of m linear equations in n variables is equivalent to a system of p linear equations in q variables, then n = q.

True, equivalent systems of linear equations always have the same number of variables (you can never reduce the number of variables and keep an equivalent system).

Suppose that the pivot rows of a matrix A are rows 1,2,...,k, and row k+1 becomes zero when applying the Gaussian elimination algorithm. Then row k + 1 must equal some linear combination of rows 1, 2, ..., k.

True, if row k + 1 becomes zero it needs to be able to become zero with only using elementary row operations and by the elementary row operations we then know that row k + 1 must a linear combination of the previous rows.

If A is a matrix with rank k, then the vectors e1, e2, ... , ek appear as columns of the reduced row echelon form of A.

True, it immediately follows from exercise 66 (which was true).

If Ax = b is consistent, then the nullity of A equals the number of free variables in the general solution of Ax = b.

True, on the bottom of page 49 it says that the number of free variables in a consistent system equals the nullity.

The rank of a matrix equals the number of pivot columns in the matrix.

True, see definition in blue rectangle on page 48.

When the forward pass of Gaussian elimination is complete, the original matrix has been transformed into one in row echelon form.

True, the forward pass transforms the matrix in the row echelon form, while the backward pass further transforms it into the reduced row echelon form.

No scaling operations are required in the forward pass of Gaussian elimination.

True, the scaling operations are required in the backward pass of the Gaussian elimination.

The third pivot position in a matrix lies in row 3.

True, the third pivot position of a matrix always lies in the third row, but in which column it lies is not in advance determined.

If R is an n x n matrix in reduced row echelon form has rank n, then R= I_n.

True, this is mentioned in the blue rectangle on the middle of page 48.

The equation Ax = b is consistent if and only if b is a linear combination of the columns of A.

True, this is stated in theorem 1.5 on page 50.