Solve Using an Inverse Matrix 2x+5y=1 , 4x+20y=6

Problem

{[2*x+5*y=1],[4*x+20*y=6])

Solution

1. Write the system in 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=[[2,5],[4,20]]

X=[[x],[y]]

B=[[1],[6]]

2. Calculate the determinant of matrix A denoted as det(A) or |A|

det(A)=(2)*(20)−(5)*(4)

det(A)=40−20

det(A)=20

3. Find the inverse matrix A(−1) using the formula A(−1)=1/det(A)*[[d,−b],[−c,a]]

A(−1)=1/20*[[20,−5],[−4,2]]

A(−1)=[[1,−0.25],[−0.2,0.1]]

4. Solve for X by multiplying the inverse matrix A(−1) by the constant vector B such that X=A(−1)*B

X=[[1,−0.25],[−0.2,0.1]]*[[1],[6]]

5. Perform the matrix multiplication to find the values of x and y

x=(1)*(1)+(−0.25)*(6)=1−1.5=−0.5

y=(−0.2)*(1)+(0.1)*(6)=−0.2+0.6=0.4

Final Answer

[[x],[y]]=[[−0.5],[0.4]]

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