Find the Nullity

Problem

A=[[1,0,2,2,1],[0,2,4,0,−2],[5,−38,6,10,10],[5,−38,6,10,10],[−5,42,−6,−10,−7]]

Solution

1. Identify the goal, which is to find the nullity of the matrix A The nullity is the dimension of the null space, calculated as the number of columns minus the rank of the matrix.

2. Simplify the matrix by removing redundant rows. Row 4 is identical to Row 3, and Row 5 is the sum of Row 3 and Row 2 multiplied by −2 (with some adjustments). We perform row operations to reach Row Echelon Form.

3. Perform row operations (R_3)→(R_3)−5*(R_1) and (R_4)→(R_4)−5*(R_1) and (R_5)→(R_5)+5*(R_1)

[[1,0,2,2,1],[0,2,4,0,−2],[0,−38,−4,0,5],[0,−38,−4,0,5],[0,42,4,0,−2]]

4. Eliminate redundant rows and simplify further. Divide (R_2) by 2

[[1,0,2,2,1],[0,1,2,0,−1],[0,−38,−4,0,5],[0,0,0,0,0],[0,42,4,0,−2]]

5. Apply (R_3)→(R_3)+38*(R_2) and (R_5)→(R_5)−42*(R_2)

[[1,0,2,2,1],[0,1,2,0,−1],[0,0,72,0,−33],[0,0,0,0,0],[0,0,−80,0,40]]

6. Observe that (R_3) and (R_5) are not multiples of each other, but they are linearly independent. For example, (R_5)→(R_5)+80/72*(R_3) will result in a non-zero value in the last column.

7. Determine the rank. There are 4 linearly independent rows (or pivots in columns 1, 2, 3, and 5). Thus, rank(A)=4

8. Calculate the nullity using the Rank-Nullity Theorem: nullity(A)=n−rank(A) where n is the number of columns.

nullity(A)=5−4=1

Final Answer

nullity(A)=1

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