### A Basic Linear Algebra Principle

##### 2. Implication of size on existence/uniqueness

By combining the numerical implications of the existence and the uniqueness on the size, we have

Ax = b has a unique solution for any b
⇒ number of rows of A = number of columns of A (A is a square matrix)
⇔ number of equations = number of variables

Conversely, assume A is an n by n matrix. Then we have (if you find the general argument too difficult, try 3 by 3 matrix first)

Ax = b has solutions for any b
⇒ All rows are pivot
⇒ Number of pivot rows is n (because there are n rows)
⇒ Number of pivot columns is n (by this equality)
⇒ All columns are pivot (because there are n columns)
⇒ The solution of a consistent system Ax = b is unique

In other words, if the number of equations is equal to the number of variables, then always existence implies uniqueness. By similar argument, we can also prove that uniqueness implies always existence.

In summary, we have the following basic principle of linear algebra.

For a square matrix A, the following are equivalent

Always Existence + Uniqueness
Ax = b has a unique solution for any b
Always Existence
Ax = b has solutions for any b
Uniqueness
The solution of a consistent system Ax = b is unique

Our discussion also tells us when the above happens from computational viewpoint.

For a square matrix A, the following are equivalent

• Ax = b has a unique solution for any b
• All rows of A are pivot
• All columns of A are pivot
• A can be row operated to become I

For the claim that A can be row operated to become I, please check out more details in this exercise.

Example The system

 x1 - x2 + 2x3 = 1 3x1 + x2 - 2x3 = 3 2x1 - x2 + 2x3 = 2

has x1 = 1, x2 = x3 = 0 as an obvious solution. It is also easy to see that x1 = 1, x2 = 2, x3 = 1 is another solution. Therefore the system has many solutions. Since the system

 x1 - x2 + 2x3 = b1 3x1 + x2 - 2x3 = b2 2x1 - x2 + 2x3 = b3

has the same coefficient matrix, by the basic principle, it does not always have solutions.

The significance of the basic principle is the following: We may consider (always) existence and the uniqueness as two complementary aspects of systems of linear equations. In general, there is no relation between the two aspects. However, in case the size is right (square coefficient matrix, or number of variables = number of equations), the two aspects are equivalent.