Find the Nullity [[1,2,2,3],[-1,-3,3,2],[2,-3,4,6]]

Problem

Nullity*[[1,2,2,3],[−1,−3,3,2],[2,−3,4,6]]

Solution

1. Set up the matrix A and prepare to find its rank using Gaussian elimination.

A=[[1,2,2,3],[−1,−3,3,2],[2,−3,4,6]]

2. Eliminate the entries in the first column below the first pivot by adding row 1 to row 2 ((R_2)+(R_1)→(R_2) and subtracting twice row 1 from row 3 ((R_3)−2*(R_1)→(R_3).

[[1,2,2,3],[0,−1,5,5],[0,−7,0,0]]

3. Eliminate the entry in the second column below the second pivot by subtracting 7 times row 2 from row 3 ((R_3)−7*(R_2)→(R_3).

[[1,2,2,3],[0,−1,5,5],[0,0,−35,−35]]

4. Identify the rank of the matrix by counting the number of non-zero rows in the row echelon form.

rank(A)=3

5. Apply the Rank-Nullity Theorem, which states that for an m×n matrix, rank(A)+nullity(A)=n where n is the number of columns.

3+nullity(A)=4

6. Solve for the nullity.

nullity(A)=4−3=1

Final Answer

nullity*[[1,2,2,3],[−1,−3,3,2],[2,−3,4,6]]=1

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