Find the Kernel

Problem

5*[[x,y],[y,z]]+2*[[2*x,−y],[3*y,−4*z]]=[[45,12−y],[33,15]]

Solution

1. Distribute the scalar constants into each matrix on the left side of the equation.

5*[[x,y],[y,z]]=[[5*x,5*y],[5*y,5*z]]

2*[[2*x,−y],[3*y,−4*z]]=[[4*x,−2*y],[6*y,−8*z]]

2. Add the two matrices on the left side by summing their corresponding entries.

[[5*x+4*x,5*y−2*y],[5*y+6*y,5*z−8*z]]=[[9*x,3*y],[11*y,−3*z]]

3. Equate the resulting matrix to the matrix on the right side to form a system of equations.

[[9*x,3*y],[11*y,−3*z]]=[[45,12−y],[33,15]]

4. Solve for x using the entry in the first row, first column.

9*x=45

x=5

5. Solve for y using the entry in the second row, first column.

11*y=33

y=3

6. Verify the value of y using the entry in the first row, second column.

3*y=12−y

3*(3)=12−3

9=9

7. Solve for z using the entry in the second row, second column.

−3*z=15

z=−5

Final Answer

5*[[x,y],[y,z]]+2*[[2*x,−y],[3*y,−4*z]]=[[45,12−y],[33,15]]⇒x=5,y=3,z=−5

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