Find the Nullity

Problem

Nullity*[[1,0.7,0.3,0],[−0.3,0.8,−0.5,0],[−0.15,−0.3,0.45,0]]

Solution

1. Identify the matrix A and its dimensions. The matrix is a 3×4 matrix, meaning it has n=4 columns.

2. Perform row operations to find the Row Echelon Form (REF). Start by multiplying the second row by 10 and the third row by 100 to clear decimals.

(R_2)→10*(R_2)

(R_3)→100*(R_3)

[[1,0.7,0.3,0],[−3,8,−5,0],[−15,−30,45,0]]

3. Eliminate the first column entries below the pivot.

(R_2)→(R_2)+3*(R_1)

(R_3)→(R_3)+15*(R_1)

[[1,0.7,0.3,0],[0,10.1,−4.1,0],[0,−19.5,49.5,0]]

4. Eliminate the second column entry below the second pivot.

(R_3)→(R_3)+19.5/10.1*(R_2)

[[1,0.7,0.3,0],[0,10.1,−4.1,0],[0,0,41.584...,0]]

5. Determine the rank of the matrix. Since there are 3 non-zero rows in the row echelon form, the rank r is 3

rank(A)=3

6. Apply the Rank-Nullity Theorem, which states that for an m×n matrix, the rank plus the nullity equals the number of columns n

rank(A)+nullity(A)=n

3+nullity(A)=4

7. Solve for the nullity.

nullity(A)=4−3

nullity(A)=1

Final Answer

Nullity*[[1,0.7,0.3,0],[−0.3,0.8,−0.5,0],[−0.15,−0.3,0.45,0]]=1

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