Find the Variables

Problem

[[1,−1,1],[0,2,1]]*[[a,d],[b,e],[c,ƒ]]=[[1,0],[0,1]]

Solution

1. Perform matrix multiplication on the left side by multiplying each row of the first matrix by each column of the second matrix.

[[(1)*(a)+(−1)*(b)+(1)*(c),(1)*(d)+(−1)*(e)+(1)*(ƒ)],[(0)*(a)+(2)*(b)+(1)*(c),(0)*(d)+(2)*(e)+(1)*(ƒ)]]=[[1,0],[0,1]]

2. Simplify the expressions within the resulting matrix to set up a system of equations.

[[a−b+c,d−e+ƒ],[2*b+c,2*e+ƒ]]=[[1,0],[0,1]]

3. Equate corresponding entries to create two independent systems of linear equations, one for variables {a,b,c} and one for {d,e,ƒ}

a−b+c=1

2*b+c=0

d−e+ƒ=0

2*e+ƒ=1

4. Solve for variables in terms of one parameter for each system, as there are more variables than equations (underdetermined system). Let c and ƒ be free variables.

c=c

2*b=−c⇒b=−c/2

a=1+b−c⇒a=1−c/2−c⇒a=1−(3*c)/2

5. Solve the second system using the same method.

ƒ=ƒ

2*e=1−ƒ⇒e=(1−ƒ)/2

d=e−ƒ⇒d=(1−ƒ)/2−ƒ⇒d=(1−3*ƒ)/2

Final Answer

[[a,d],[b,e],[c,ƒ]]=[[1−(3*c)/2,(1−3*ƒ)/2],[−c/2,(1−ƒ)/2],[c,ƒ]]

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