Solve Using an Inverse Matrix x-z+3y=4 , z=3y , y-x=5z

Problem

{[x−z+3*y=4],[z=3*y],[y−x=5*z])

Solution

1. Rearrange the equations into standard form a*x+b*y+c*z=d so that variables are aligned.

x+3*y−z=4

0*x−3*y+z=0

−x+y−5*z=0

2. Write the system as a matrix equation A*X=B

A=[[1,3,−1],[0,−3,1],[−1,1,−5]],X=[[x],[y],[z]],B=[[4],[0],[0]]

3. Calculate the determinant of matrix A using the first column.

det(A)=1*((−3)*(−5)−(1)*(1))−0+(−1)*((3)*(1)−(−1)*(−3))

det(A)=1*(15−1)−1*(3−3)

det(A)=14

4. Find the adjugate matrix adj(A) by calculating the transpose of the cofactor matrix.

(C_11)=14,(C_12)=−1,(C_13)=−3

(C_21)=14,(C_22)=−6,(C_23)=−4

(C_31)=0,(C_32)=−1,(C_33)=−3

adj(A)=[[14,14,0],[−1,−6,−1],[−3,−4,−3]]

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

A(−1)=1/14*[[14,14,0],[−1,−6,−1],[−3,−4,−3]]

6. Solve for X by multiplying A(−1) and B

X=1/14*[[14,14,0],[−1,−6,−1],[−3,−4,−3]]*[[4],[0],[0]]

x=1/14*(14⋅4+14⋅0+0⋅0)=4

y=1/14*(−1⋅4−6⋅0−1⋅0)=−4/14=−2/7

z=1/14*(−3⋅4−4⋅0−3⋅0)=−12/14=−6/7

Final Answer

[[x],[y],[z]]=[[4],[−2/7],[−6/7]]

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