Find the Kernel

Problem

[[6,−2,−4,4],[3,−3,−6,1],[−12,8,21,−8],[−6,0,−10,7]]*[[x],[y],[z],[w]]=[[2],[−4],[8],[−43]]

Solution

1. Identify the task. The kernel (or null space) of a matrix A is the set of all vectors v such that A*v=0 The provided non-homogeneous equation A*x=b is irrelevant to finding the kernel of the matrix itself.

2. Set up the augmented matrix for the homogeneous system A*x=0 to find the kernel.

[[6,−2,−4,4,0],[3,−3,−6,1,0],[−12,8,21,−8,0],[−6,0,−10,7,0]]

3. Perform row operations to reach row echelon form. Divide the first row by 2 and swap with the second row to create a leading 1

[[3,−3,−6,1,0],[6,−2,−4,4,0],[−12,8,21,−8,0],[−6,0,−10,7,0]]

4. Eliminate entries below the first pivot. (R_2)→(R_2)−2*(R_1) (R_3)→(R_3)+4*(R_1) and (R_4)→(R_4)+2*(R_1)

[[3,−3,−6,1,0],[0,4,8,2,0],[0,−4,−3,−4,0],[0,−6,−22,9,0]]

5. Eliminate entries below the second pivot. (R_3)→(R_3)+(R_2) and (R_4)→(R_4)+1.5*(R_2)

[[3,−3,−6,1,0],[0,4,8,2,0],[0,0,5,−2,0],[0,0,−10,12,0]]

6. Eliminate the entry below the third pivot. (R_4)→(R_4)+2*(R_3)

[[3,−3,−6,1,0],[0,4,8,2,0],[0,0,5,−2,0],[0,0,0,8,0]]

7. Analyze the resulting upper triangular matrix. Since there are four pivots (on the diagonal) for a 4×4 matrix, the matrix is non-singular (invertible).

8. Conclude that the only solution to the homogeneous system A*x=0 is the trivial solution.

Final Answer

Kernel={[0],[0],[0],[0]}

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