Understanding Matrix Rank, Basic and Free Variables

This lesson explains how to determine the rank, basic variables, and free variables of a coefficient matrix. It covers the concepts of leading entries, pivot positions, and the relationship between the rank and the number of parameters in a homogeneous system. Two examples are provided to illustrate these concepts, showing how to identify leading entries and determine basic and free variables in different matrix forms.

What is the rank of a matrix?

• The number of leading entries in echelon form

• The number of rows in the matrix

• The number of columns in the matrix

• The number of zero entries in the matrix

In a homogeneous system, if a matrix has n columns and rank r, how many free variables are there?

• n + r

• n - r

• r + n

• r - n

What is the first step in determining the rank of a matrix using row echelon form?

• Counting the number of columns

• Counting the number of rows

• Identifying the leading entries

• Identifying the zero entries

In the first example, which columns are identified as pivot columns?

• Column two and column four

• Column one and column three

• Column two and column three

• Column one and column two

In the first example, which variable is identified as a free variable?

• x sub 1

• x sub 2

• x sub 4

• x sub 3

In the second example, how many leading entries are there?

• Two

• Five

• Four

• Three

Which columns are pivot columns in the second example?

• Column two, column three, and column four

• Column one, column four, and column five

• Column one, column two, and column three

• Column one, column three, and column four

In the second example, which variables are free variables?

• x sub 2 and x sub 5

• x sub 4 and x sub 5

• x sub 1 and x sub 2

• x sub 3 and x sub 4

What is the relationship between the number of columns, rank, and free variables in a matrix?

• Number of columns = rank - free variables

• Number of columns = free variables - rank

• Number of columns = rank * free variables

• Number of columns = rank + free variables

What does a solution with two parameters indicate in the context of a homogeneous system?

• Two basic variables

• Two free variables

• Two leading entries

• Two pivot columns