Solve Using an Inverse Matrix 2b+c+z=11 , 3b+4c+z=19 , 3b+6c+5z=43

Problem

{[2*b+c+z=11],[3*b+4*c+z=19],[3*b+6*c+5*z=43])

Solution

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

A=[[2,1,1],[3,4,1],[3,6,5]]

X=[[b],[c],[z]]

B=[[11],[19],[43]]

2. Calculate the determinant of matrix A to ensure it is invertible.

det(A)=2*(4⋅5−1⋅6)−1*(3⋅5−1⋅3)+1*(3⋅6−4⋅3)

det(A)=2*(14)−1*(12)+1*(6)

det(A)=28−12+6=22

3. Find the adjugate matrix adj(A) by calculating the transpose of the matrix of cofactors.

(C_11)=14,(C_12)=−12,(C_13)=6

(C_21)=1,(C_22)=7,(C_23)=−9

(C_31)=−3,(C_32)=1,(C_33)=5

adj(A)=[[14,1,−3],[−12,7,1],[6,−9,5]]

4. Determine the inverse matrix A(−1) using the formula A(−1)=1/det(A)*adj(A)

A(−1)=1/22*[[14,1,−3],[−12,7,1],[6,−9,5]]

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

X=1/22*[[14,1,−3],[−12,7,1],[6,−9,5]]*[[11],[19],[43]]

X=1/22*[[14*(11)+1*(19)−3*(43)],[−12*(11)+7*(19)+1*(43)],[6*(11)−9*(19)+5*(43)]]

X=1/22*[[154+19−129],[−132+133+43],[66−171+215]]

X=1/22*[[44],[44],[110]]

6. Simplify the vector to find the values of b c and z

b=44/22=2

c=44/22=2

z=110/22=5

Final Answer

[[b],[c],[z]]=[[2],[2],[5]]

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