Eco 134: Applied Mathematics - Matrix Algebra

ECO 134: Applied Mathematics I [Course Instructor: Saima Khan(Sai)]

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Matrix Algebra: Does a Unique Solution Exist? he question ‘ Does a unique solution exist' may be asked in different ways. Essentially, the concepts you use for your answer is always the same – however the way in which you phrase your answer will differ based on what the question is asking for. Example 1: You are given the following set of linear equations. Does a unique solution exist? So the coefficient matrix here is:

For a unique solution to exist, the following conditions must hold:

Necessary Condition:

the coefficient matrix, A, must be square

Since, A is a 2x2 matrix --> it is a square matrix.

Hence the necessary condition holdsThis implies that there are 2 equations and 2 unknown variables

Thus there is a possibility that unique solution mayexists

Sufficient Condition:

the rows of the coefficient matrix must be linearly independent.

Since |A|≠0, the rows are linearly independent

This implies that the equations we are dealing with are functionally independent and consistent

Hence now we are confirmed that a unique solutiondoes exist.

ECO 134: Applied Mathematics I [Course Instructor: Saima Khan(Sai)]

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Example 2: The same question may be asked in a slightly different way: Q: Based on the rank that you just found, explain whether a unique solution exists or not? In this case, recall that we are still working with the same set of linear equations presented above. So how is the answer different? Well the concept is still the same – to get a unique solution the same necessary and sufficient conditions must hold. However, the difference is only in the way you phrase your answer. Note that the addition/change that has been made is shown in italics and underlined form.

Note that we used the term full rank in our answer – what do we mean by full rank? Well rank is the number of independent rows of columns that a matrix has. When we say that a matrix has full rank, it means all the rows/columns are independent. So, if we are dealing with a 2x2 matrix, if its rank is 2, then we say it is a full rank matrix. If we are dealing with a 3x3 matrix, then if its rank is 3, we say it is a full rank matrix.

For a unique solution to exist, the following conditions must hold:

Necessary Condition:

the coefficient matrix,