Solve Using an Inverse Matrix x-y=5 , x+2y=11 , 2x+y=16

Problem

{[x−y=5],[x+2*y=11],[2*x+y=16])

Solution

1. Write the system in matrix form A*X=B

A=([1,−1],[1,2],[2,1])

X=([x],[y])

B=([5],[11],[16])

2. Recognize that since A is not a square matrix, we use the left inverse (pseudo-inverse) method by multiplying both sides by the transpose A^T

A^T*A*X=A^T*B

3. Calculate the product A^T*A

A^T*A=([1,1,2],[−1,2,1])*([1,−1],[1,2],[2,1])

A^T*A=([1+1+4,−1+2+2],[−1+2+2,1+4+1])

A^T*A=([6,3],[3,6])

4. Calculate the product A^T*B

A^T*B=([1,1,2],[−1,2,1])*([5],[11],[16])

A^T*B=([5+11+32],[−5+22+16])

A^T*B=([48],[33])

5. Find the inverse of A^T*A

det(A^T*A)=(6)*(6)−(3)*(3)=27

(A^T*A)^(−1)=1/27*([6,−3],[−3,6])

6. Solve for X using X=(A^T*A)^(−1)*A^T*B

X=1/27*([6,−3],[−3,6])*([48],[33])

X=1/27*([288−99],[−144+198])

X=1/27*([189],[54])

X=([7],[2])

7. Verify the solution in the original equations.

7−2=5

7+2*(2)=11

2*(7)+2=16

Final Answer

([x],[y])=([7],[2])

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