Rank And Nullity Calculator and Formula

A Rank and Nullity Calculator is a tool (usually a software program or an online calculator) that computes the rank and nullity of a given matrix.

It automates the process of finding the row echelon form, counting the non-zero rows, and applying the nullity formula.

How to find rank and nullity of a matrix?

To find the rank and nullity of a matrix, follow these steps:

1. Reduce the matrix to row echelon form (REF) using row operations (row swapping, row scaling, and row addition).
2. Count the number of non-zero rows in the REF. This gives you the rank of the matrix.
3. The nullity is the dimension of the null space, which is given by the formula: nullity = number of columns - rank

Rank And Nullity Calculator

Consider the following matrix:

A = [ 1  2  3 ]
    [ 2  4  6 ]

To find the rank and nullity of this matrix, we can use the following steps:

Step 1: Convert the matrix to row echelon form (REF) using row operations.

REF(A) = [ 1  2  3 ]
         [ 0  0  0 ]

Step 2: Count the number of non-zero rows in the REF to find the rank.

In this case, there is 1 non-zero row, so the rank of the matrix A is 1.

Step 3: Apply the nullity formula to find the nullity.

nullity = number of columns - rank
        = 3 - 1
        = 2

Therefore, the rank of matrix A is 1, and the nullity is 2.

Let’s try another example:

Consider the matrix:

B = [ 1  2  1  3 ]
    [ 2  4  2  6 ]
    [ 3  6  3  9 ]

Step 1: Convert the matrix to REF.

REF(B) = [ 1  2  1  3 ]
         [ 0  0  0  0 ]
         [ 0  0  0  0 ]

Step 2: Count the number of non-zero rows in the REF to find the rank.

In this case, there is 1 non-zero row, so the rank of matrix B is 1.

Step 3: Apply the nullity formula to find the nullity.

nullity = number of columns - rank
        = 4 - 1
        = 3

Therefore, the rank of matrix B is 1, and the nullity is 3.

These examples demonstrate how a Rank and Nullity Calculator can be used to find the rank and nullity of matrices by converting them to row echelon form, counting non-zero rows, and applying the nullity formula.

How Rank And Nullity Calculator Works

A Rank and Nullity Calculator typically works as follows:

1. Input Matrix: The user inputs the matrix for which they want to find the rank and nullity.
2. Row Echelon Form: The calculator performs row operations to convert the input matrix into its row echelon form.
3. Rank Calculation: The calculator counts the number of non-zero rows in the REF, which gives the rank of the matrix.
4. Nullity Calculation: The calculator applies the formula nullity = number of columns - rank to find the nullity.
5. Output: The calculator displays the rank and nullity of the input matrix.

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Rank And Nullity Formula

The formula for calculating the rank and nullity of a matrix is:

• Rank = The number of non-zero rows in the row echelon form (REF) of the matrix.
• Nullity = number of columns - rank

What is Rank And Nullity?

Rank and Nullity are important concepts in linear algebra that provide information about the fundamental properties of a matrix.

• Rank: The rank of a matrix is the dimension of the column space or the number of linearly independent columns in the matrix. It represents the maximum number of linearly independent columns or rows in the matrix.
• Nullity: The nullity of a matrix is the dimension of the null space or the number of linearly independent vectors in the null space of the matrix. It represents the number of non-trivial solutions to the homogeneous system of linear equations represented by the matrix.

Rank and nullity are related by the following equation:

rank + nullity = number of columns

This relationship is known as the Rank-Nullity Theorem, which states that the sum of the rank and nullity of a matrix is always equal to the number of columns in the matrix.

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