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**The Rank of a Matrix**

You can think of an *r* x *c* matrix as a set of *r* row vectors, each having *c* elements; or you can think of it as a set of *c* column vectors, each having *r* elements.

The **rank** of a matrix is defined as (a) the maximum number of linearly independent *column* vectors in the matrix or (b) the maximum number of linearly independent *row* vectors in the matrix. Both definitions are equivalent.

For an *r* x *c* matrix,

- If
*r*is less than*c*, then the maximum rank of the matrix is*r*. - If
*r*is greater than*c*, then the maximum rank of the matrix is*c*.

The rank of a matrix would be zero only if the matrix had no elements. If a matrix had even one element, its minimum rank would be one.

**How to Find Matrix Rank**

In this section, we describe a method for finding the rank of any matrix. This method assumes familiarity withechelon matrices and echelon transformations.

The maximum number of linearly independent vectors in a matrix is equal to the number of non-zero rows in its row echelon matrix. Therefore, to find the rank of a matrix, we simply transform the matrix to its row echelon form and count the number of non-zero rows.

Consider matrix **A** and its row echelon matrix, **A _{ref}**. Previously, we showed how to find the row echelon form for matrix

**Full Rank Matrices**

When all of the vectors in a matrix are linearly independent, the matrix is said to be **full rank**. Consider the matrices **A** and **B** below.

Notice that row 2 of matrix **A** is a scalar multiple of row 1; that is, row 2 is equal to twice row 1. Therefore, rows 1 and 2 are linearly dependent. Matrix **A** has only one linearly independent row, so its rank is 1. Hence, matrix **A** is not full rank.

Now, look at matrix **B**. All of its rows are linearly independent, so the rank of matrix **B** is 3. Matrix **B** is full rank.

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