Find the Kernel a=[[3,-1],[2,3]]b=[[4,1],[2,-3]]

Problem

ker*([[3,−1],[2,3]]*[[4,1],[2,−3]])

Solution

1. Multiply the two matrices A and B to find the resulting matrix M

M=[[3,−1],[2,3]]*[[4,1],[2,−3]]

M=[[(3)*(4)+(−1)*(2),(3)*(1)+(−1)*(−3)],[(2)*(4)+(3)*(2),(2)*(1)+(3)*(−3)]]

M=[[10,6],[14,−7]]

2. Set up the homogeneous system M*x=0 to find the kernel.

[[10,6],[14,−7]]*[[x],[y]]=[[0],[0]]

3. Calculate the determinant of M to check if the kernel is trivial.

det(M)=(10)*(−7)−(6)*(14)

det(M)=−70−84

det(M)=−154

4. Conclude that since the determinant is non-zero, the matrix is invertible.

det(M)≠0⇒M*x=0* has only the trivial solution

x=[[0],[0]]

Final Answer

ker*([[3,−1],[2,3]]*[[4,1],[2,−3]])={[0],[0]}

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