Table of Contents
Introduction to Rank of a Matrix
A matrix is a rectangular array of numbers arranged in rows and columns. The rank of a matrix is the number of linearly independent rows or columns in the matrix. A matrix is rank deficient if its rank is less than the number of rows or columns in the matrix.
The rank of a Matrix Definition
The rank of a matrix is the dimension of the largest vector space spanned by its columns.
Nullity of a Matrix
A matrix is said to be null if its determinant is zero. The nullity of a matrix is the dimension of the vector space spanned by the columns of the matrix.
Properties of the Rank of the Matrix:
The rank of a matrix is the number of linearly independent rows or columns in the matrix.
How to Find the Rank of the Matrix?
To find the rank of the matrix, one can use the following formula:
rank(A) = the number of linearly independent columns in A
rank(A) = the number of linearly independent rows in A
To Calculate Rank of Matrix There are Two Methods:
1.Using determinant
2.Using inverse
Rank of Matrix Using Determinant:
1.First, calculate the determinant of the matrix.
2.Next, divide the determinant by the number of columns in the matrix.
3.Finally, take the square root of the result. This will be the rank of the matrix.
Rank of Matrix Using Inverse:
1.First, calculate the inverse of the matrix.
2.Next, multiply the inverse of the matrix by the original matrix.
3.Finally, take the square root of the result. This will be the rank of the matrix.
Steps to Find the Rank of the Matrix by Minor Method:
1. Find the rank of the matrix A by using the determinant method.
2. Find the rank of the matrix A by using the determinant method, again.
3. Find the rank of the matrix A by using the minor method.
4. Compare the ranks found in steps 2 and 3. The rank of A by the minor method is smaller than the rank of A by the determinant method. Therefore, the rank of A by the minor method is the rank of A.
Steps to Find the Rank of the Matrix by Echelon Form:
1. Write the matrix in echelon form
2. Find the rank of the matrix
3. The rank of the matrix is
The rank of the matrix is 3.
