Solve Using an Inverse Matrix x-2y=3 , -2x+4y=6

Problem

{[x−2*y=3],[−2*x+4*y=6])

Solution

1. Write the system in the matrix form A*X=B where A is the coefficient matrix, X is the variable column vector, and B is the constant column vector.

A=[[1,−2],[−2,4]]

X=[[x],[y]]

B=[[3],[6]]

2. Calculate the determinant of the coefficient matrix A using the formula det(A)=a*d−b*c

det(A)=(1)*(4)−(−2)*(−2)

det(A)=4−4

det(A)=0

3. Determine the existence of the inverse matrix. Since the determinant is zero, the matrix A is singular and does not have an inverse.

det(A)=0⇒A(−1)* does not exist

4. Analyze the equations to check for consistency. Multiply the first equation by −2

−2*(x−2*y)=−2*(3)

−2*x+4*y=−6

5. Compare the results with the second equation of the system. The second equation is −2*x+4*y=6

−6≠6

6. Conclude the solution type. Because the left sides are identical but the right sides are different, the lines are parallel and never intersect.

Final Answer

No solution

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