Solve Using an Inverse Matrix x+2y=1 , 4x+5y=13

Problem

{[x+2*y=1],[4*x+5*y=13])

Solution

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

A=[[1,2],[4,5]]

X=[[x],[y]]

B=[[1],[13]]

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

det(A)=(1)*(5)−(2)*(4)

det(A)=5−8=−3

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

A(−1)=1/(−3)*[[5,−2],[−4,1]]

A(−1)=[[−5/3,2/3],[4/3,−1/3]]

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

X=[[−5/3,2/3],[4/3,−1/3]]*[[1],[13]]

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

x=(−5/3)*(1)+(2/3)*(13)=−5/3+26/3=21/3=7

y=(4/3)*(1)+(−1/3)*(13)=4/3−13/3=−9/3=−3

Final Answer

[[x],[y]]=[[7],[−3]]

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