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Nullity of a Matrix

  • Last Updated : 03 Jun, 2019

Prerequisite – Mathematics | System of Linear Equations
Let A be a matrix. Since, number of non-zero rows in the row reduced form of a matrix A is called the rank of A, denoted as rank(A) and Nullity is the complement to the rank of a matrix .Please go through the Prerequisite first and read the rank topic, then come to this topic.

Therefore, Nullity of a matrix is calculated from rank of the matrix using the following steps:Let A[m*n] matrix, then:

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  1. Calculate rank (r) of the Matrix.
  2. Use The Rank Plus Nullity Theorem, it says
    Nullity + rank = number of columns (n)

    Therefore, you will be able to calculate nullity as



    Nullity = no. of columns(n) - rank(r) 

Consider the examples:

Example-1:

Input: mat[][] = {{10,   20,   10},
                 {20,   40,   20},
                 {30,   50,   0}}

Output: Rank is 2 and hence Nullity is 1 

Explanation: Ist and IInd rows are linearly dependent. But Ist and 3rd or IInd and IIIrd are independent, so Rank is 2 and hence Nullity is (3-2) = 1.

Example-2:

Input: mat[][] = {{1,   2,   1},
                 {2,   3,   1},
                 {1,   1,   2}}

Output: Rank is 3 and hence Nullity is 0 

Explanation: Ist and IInd and IIIrd rows are linearly dependent, so Rank is 3 and hence Nullity is (3-3) = 0.

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