Find the Kernel

Problem

[[9,−36,30],[−36,192,−180],[30,−180,180]]*[[x],[y],[z]]=[[15],[−48],[30]]

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 Although an inhomogeneous equation A*x=b is provided, finding the kernel refers to solving the homogeneous system A*x=0

2. Set up the augmented matrix for the homogeneous system A*x=0 using the coefficients of the matrix:

[[9,−36,30,0],[−36,192,−180,0],[30,−180,180,0]]

3. Simplify the first row by dividing by 3

[[3,−12,10,0],[−36,192,−180,0],[30,−180,180,0]]

4. Eliminate the first column entries below the pivot by performing (R_2)←(R_2)+12*(R_1) and (R_3)←(R_3)−10*(R_1)

[[3,−12,10,0],[0,48,−60,0],[0,−60,80,0]]

5. Simplify the second and third rows by dividing (R_2) by 12 and (R_3) by 20

[[3,−12,10,0],[0,4,−5,0],[0,−3,4,0]]

6. Eliminate the entry in the third row, second column by performing (R_3)←(R_3)+3/4*(R_2)

[[3,−12,10,0],[0,4,−5,0],[0,0,1/4,0]]

7. Determine the rank. Since there are three pivots (the diagonal entries are non-zero), the matrix is non-singular (invertible).

8. Solve the homogeneous system. Since the matrix is invertible, the only solution to A*x=0 is the trivial solution.

Final Answer

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

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