Rank of Matrix

# Rank of Matrix | Engineering Mathematics - Civil Engineering (CE) PDF Download

## 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 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.

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.

The document Rank of Matrix | Engineering Mathematics - Civil Engineering (CE) is a part of the Civil Engineering (CE) Course Engineering Mathematics.
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## FAQs on Rank of Matrix - Engineering Mathematics - Civil Engineering (CE)

 1. What is the rank of a matrix?
Ans. The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. It represents the dimension of the vector space spanned by the rows or columns of the matrix.
 2. How is the rank of a matrix calculated?
Ans. The rank of a matrix can be calculated by performing row reduction operations to bring the matrix into its row echelon form or reduced row echelon form. The number of non-zero rows in the resulting form is equal to the rank of the matrix.
 3. How does the rank of a matrix relate to its invertibility?
Ans. A square matrix is invertible if and only if its rank is equal to the number of rows (or columns) in the matrix. In other words, a matrix is invertible if it has full rank. If the rank is less than the number of rows, the matrix is singular and does not have an inverse.
 4. Can the rank of a matrix be greater than its dimensions?
Ans. 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 the rank is equal to the minimum, the matrix is said to have full rank.
 5. How can the rank of a matrix be used in solving systems of linear equations?
Ans. The rank of a coefficient matrix in a system of linear equations can provide information about the solvability and uniqueness of the system. If the rank of the coefficient matrix is equal to the rank of the augmented matrix, the system has a unique solution. If the rank is less than the rank of the augmented matrix, the system either has no solution or infinitely many solutions.

## Engineering Mathematics

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