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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 Rank of Matrix

In this section, we describe a method for finding the rank of any matrix. This method assumes familiarity with echelon 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, Aref. Previously, we showed how to find the row echelon form for matrix A.

Because the row echelon form Aref has two non-zero rows, we know that matrix A has two independent row vectors; and we know that the rank of matrix A is 2.

You can verify that this is correct. Row 1 and Row 2 of matrix A are linearly independent. However, Row 3 is a linear combination of Rows 1 and 2. Specifically, Row 3 = 3*( Row 1 ) + 2*( Row 2). Therefore, matrix A has only two independent row vectors.

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