Rank of Matrix: The rank of a matrix is defined as:
1 The maximum number of linearly independent column vectors in the matrix
2 The maximum number of linearly independent row vectors in the matrix.
For any matrix with dimensions of m*n
1 If m is less than n then maximum rank of a matrix is n
2 If m is greater than n then maximum rank of a matrix is m
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 determine Rank of a Matrix: In this section, we describe a method for finding the rank of any matrix.
The maximum number of linearly independent vectors in a matrix is equal to the number of non-zero rows in itsrow 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.
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.