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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, Aref. Previously, we showed how to find the row echelon form for matrix A.
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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FAQs on Rank of Matrices - Matrix Algebra, CSIR-NET Mathematical Sciences - Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
| 1. What is the rank of a matrix? |  |
| 2. How is the rank of a matrix determined? | |
Answer: The rank of a matrix can be determined by performing row operations to bring the matrix into its row-echelon form or reduced row-echelon form. The number of non-zero rows in the row-echelon form or reduced row-echelon form is equal to the rank of the matrix.
| 3. What does a rank-deficient matrix mean? | |
Answer: A rank-deficient matrix is a matrix that does not have full rank. In other words, it has at least one row or column that is a linear combination of other rows or columns in the matrix. This means that the matrix does not span the full vector space.
| 4. How does the rank of a matrix affect its solutions? | |
Answer: The rank of a matrix affects the number of solutions to a system of linear equations. If the rank of the coefficient matrix is equal to the rank of the augmented matrix, then the system has a unique solution. If the rank of the coefficient matrix is less than the rank of the augmented matrix, then the system has infinitely many solutions. If the rank of the coefficient matrix is greater than the rank of the augmented matrix, then the system has no solution.
| 5. Can the rank of a matrix be greater than its dimensions? | |
Answer: No, the rank of a matrix cannot be greater than its dimensions. The rank of a matrix is always less than or equal to the minimum of the number of rows and columns in the matrix. If a matrix has more rows than columns, its rank cannot be greater than the number of columns, and vice versa.
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