Find the Kernel A=[[-1,3],[5,4]]B=[[0,-2,5],[1,-3,2]]

Problem

A=[[−1,3],[5,4]],B=[[0,−2,5],[1,−3,2]],Find the Kernel of *A*B

Solution

1. Multiply the matrices A and B to find the transformation matrix M=A*B

M=[[(−1)*(0)+(3)*(1),(−1)*(−2)+(3)*(−3),(−1)*(5)+(3)*(2)],[(5)*(0)+(4)*(1),(5)*(−2)+(4)*(−3),(5)*(5)+(4)*(2)]]

M=[[3,−7,1],[4,−22,33]]

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

[[3,−7,1],[4,−22,33]]*[[x],[y],[z]]=[[0],[0]]

3. Perform row reduction on the augmented matrix to find the relationship between variables.

[[3,−7,1],[4,−22,33]]⇒(R_1)→1/3*(R_1)⇒[[1,−7/3,1/3],[4,−22,33]]

(R_2)→(R_2)−4*(R_1)⇒[[1,−7/3,1/3],[0,−38/3,95/3]]

(R_2)→−3/38*(R_2)⇒[[1,−7/3,1/3],[0,1,−5/2]]

(R_1)→(R_1)+7/3*(R_2)⇒[[1,0,−11/2],[0,1,−5/2]]

4. Express the variables in terms of the free variable z

x−11/2*z=0⇒x=11/2*z

y−5/2*z=0⇒y=5/2*z

5. Write the solution in vector form by choosing a convenient value for z (e.g., z=2.

x=z*[[11/2],[5/2],[1]]⇒x=c*[[11],[5],[2]]

Final Answer

ker*(A*B)=span*{[11],[5],[2]}

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