Solutions of Linear Equations

An m × n matrix is said to be in (reduced) row echelon form

when it satisfies the following:

1. all zero rows are below the non-zero rows.

2. the leading (first non-zero) entry in the non-zero row is equal

to 1.

3. the leading entry of a row lies in a column to the right of the

leading entries of the preceding rows.

4. ++ each leading entry is the only non-zero entry in its column.

Every non-zero m× n matrix is row equivalent to a unique ma-

trix in reduced row echelon form. The equivalence is done using

elementary row operations:

1. interchanging 2 rows (Pij)

2. multiplying a row by a non-zero scalar (Mi(c))

3. adding to a row a multiple of another row (Aij(c))

Examples:

1. A =

1 2 3

2 5 7

−2 −4 −5

4. M =

1 −2 0 2

2 −3 −1 5

1 3 2 5

1 1 0 2

2. B =

3 −2 1

−4 1 −1

2 0 1

3. C =

1 0 3

−3 1 4

4 2 2

5 −1 5

1

The Gaussian Elimination Technique

Suppose we have a linear system AX = B. Then the solution

of the system is determined as follows:

1. Form the augmented matrix [A...B].

2. Transform this augmented matrix to row echelon form using

the 3 elementary row operations.

3. Each non-zero row determines a solution to one of the un-

knowns (via back substitution).

The Gauss-Jordan Reduction Technique

Suppose we have a linear system AX = B. Then the solution

of the system is determined as follows:

1. Form the augmented matrix [A...B].

2. Transform this augmented matrix to reduced row echelon form

using the 3 elementary row operations.

3. Each non-zero row determines a solution to one of the un-

knowns (via back substitution).

Examples:

1.

x + 2y + 3z = 9

2x− y + z = 8

3x − z = 3

2.

x + y + 2z = −1

x− 2y + z = −5

3x + y + z = 3

3.

x + y + z + w = 6

2x + y − z = 3

3x + y + 2w = 6

2

Rank of a Matrix

The number of non-zero rows in the row echelon form or the

reduced row echelon form of a matrix is called its rank.

Theorem: Let AX = B be a nonhomogeneous system of m

equations in n unknowns. Then one and only one of the following

holds:

1. rank A < rank [A...B] −→ system is inconsistent.

2. rank A = rank [A...B] = n −→ system has a unique solution.

3. rank A = rank [A...B] < n −→ system has more than one

solution.

Theorem: Let AX = 0 be a homogeneous system of m equations

in n unknowns. Then the system always has a solution, the trivial

solution X = 0. Moreover, if

1. rank A = n −→ solution is unique, and it is trivial.

2. rank A < n −→ solution is not unique.

Corollary: If A is m × n and m < n, then AX = 0 has a

non-trivial solution.

Exercise: Find the value of a for which the following system will

be consistent:x− 3y + 2z = 4

2x + y − z = 1

3x− 2y + z = a

Exercise: Find all 3 × 1 vectors B for which AX = B can be

solved, and find the corresponding general solution:

A =

4 −1 2 6

−1 5 −1 −3

3 4 1 3

3