Find the Kernel A=[[1,4],[3,6]]B=[[5,6],[2,5]]

Problem

A=[[1,4],[3,6]],B=[[5,6],[2,5]]

Solution

1. Identify the task as finding the kernel (null space) of the matrix product A*B

2. Multiply the matrices A and B to find the matrix C=A*B

C=[[1,4],[3,6]]*[[5,6],[2,5]]

C=[[(1)*(5)+(4)*(2),(1)*(6)+(4)*(5)],[(3)*(5)+(6)*(2),(3)*(6)+(6)*(5)]]

C=[[13,26],[27,48]]

3. Set up the equation C*x=0 to find the kernel, where x=[[(x_1)],[(x_2)]]

[[13,26],[27,48]]*[[(x_1)],[(x_2)]]=[[0],[0]]

4. Row reduce the augmented matrix to solve the system of linear equations.

[[13,26,0],[27,48,0]]

Divide the first row by 13

[[1,2,0],[27,48,0]]

Subtract 27 times the first row from the second row:

[[1,2,0],[0,−6,0]]

Divide the second row by −6

[[1,2,0],[0,1,0]]

Subtract 2 times the second row from the first row:

[[1,0,0],[0,1,0]]

5. Conclude that since the reduced row echelon form is the identity matrix, the only solution is the trivial solution.

(x_1)=0

(x_2)=0

Final Answer

ker*(A*B)={[0],[0]}

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