Find the Kernel

Problem

ker*(([1,3,5],[3,4,6],[5,6,7])*([6,0,0],[0,5,0],[0,0,3]))

Solution

1. Multiply the two matrices A and B to find the transformation matrix M Since B is a diagonal matrix, each column of A is scaled by the corresponding diagonal element of B

M=([1*(6),3*(5),5*(3)],[3*(6),4*(5),6*(3)],[5*(6),6*(5),7*(3)])

M=([6,15,15],[18,20,18],[30,30,21])

2. Set up the homogeneous system M*x=0 to find the kernel (null space).

([6,15,15],[18,20,18],[30,30,21])*([x],[y],[z])=([0],[0],[0])

3. Perform row reduction on the augmented matrix to find the row echelon form. Divide the first row by 3.

([2,5,5],[18,20,18],[30,30,21])

4. Eliminate the first column entries below the pivot. Subtract 9 times the first row from the second, and 15 times the first row from the third.

([2,5,5],[0,−25,−27],[0,−45,−54])

5. Simplify the third row by dividing by 9.

([2,5,5],[0,−25,−27],[0,−5,−6])

6. Eliminate the second column entry in the third row. Subtract 1/5 of the second row from the third row.

([2,5,5],[0,−25,−27],[0,0,−0.6])

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

det(M)≠0

8. Conclude that the only solution to the homogeneous system is the trivial solution.

x=([0],[0],[0])

Final Answer

ker*(([1,3,5],[3,4,6],[5,6,7])*([6,0,0],[0,5,0],[0,0,3]))={[0],[0],[0]}

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