Find the Variables

Problem

[[2,1,0],[r,0,1],[1,2,−2]]*[[a],[b],[c]]=[[0],[0],[0]]

Solution

1. Perform matrix multiplication to convert the matrix equation into a system of linear equations.

2*a+b=0

r*a+c=0

a+2*b−2*c=0

2. Solve for b in terms of a using the first equation.

b=−2*a

3. Solve for c in terms of a and r using the second equation.

c=−r*a

4. Substitute the expressions for b and c into the third equation to find the relationship between a and r

a+2*(−2*a)−2*(−r*a)=0

a−4*a+2*r*a=0

−3*a+2*r*a=0

5. Factor the equation to determine the possible values for the variables.

a*(2*r−3)=0

6. Analyze the cases for the solution. If 2*r−3≠0 (meaning r≠3/2, then a must be 0 which leads to the trivial solution a=0,b=0,c=0 If r=3/2 there are infinitely many solutions where a is a free parameter.

7. Express the variables in terms of the parameter a for the case where r=3/2

b=−2*a

c=−3/2*a

Final Answer

[[a],[b],[c]]=[[a],[−2*a],[−3/2*a* for *r=3/2]]* or *[[0],[0],[0* for *r≠3/2]]

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