Find the Kernel

Problem

[[a+2,3*z+1,5*m],[4*k,0,3]]+[[3*a,2*z,5*m],[2*k,5,6]]=[[10,−14,80],[10,5,9]]

Solution

1. Perform matrix addition by adding corresponding entries from the two matrices on the left side of the equation.

[[(a+2)+3*a,(3*z+1)+2*z,5*m+5*m],[4*k+2*k,0+5,3+6]]=[[10,−14,80],[10,5,9]]

2. Simplify the expressions within the resulting matrix to prepare for entry-wise comparison.

[[4*a+2,5*z+1,10*m],[6*k,5,9]]=[[10,−14,80],[10,5,9]]

3. Set up equations for each variable by equating corresponding entries from the two matrices.

4*a+2=10

5*z+1=−14

10*m=80

6*k=10

4. Solve for a by subtracting 2 and dividing by 4.

4*a=8

a=2

5. Solve for z by subtracting 1 and dividing by 5.

5*z=−15

z=−3

6. Solve for m by dividing by 10.

m=8

7. Solve for k by dividing by 6 and simplifying the fraction.

k=10/6

k=5/3

Final Answer

[[a],[z],[m],[k]]=[[2],[−3],[8],[5/3]]

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