Solve Using an Inverse Matrix -x-4y-7z=-4 , x-7y-z=-7 , -x+6z=0

Problem

{[−x−4*y−7*z=−4],[x−7*y−z=−7],[−x+6*z=0])

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=[[−1,−4,−7],[1,−7,−1],[−1,0,6]]

X=[[x],[y],[z]]

B=[[−4],[−7],[0]]

2. Calculate the determinant of matrix A using expansion along the third row.

det(A)=−1*((−4)*(−1)−(−7)*(−7))+0−6*((−1)*(−7)−(−4)*(1))

det(A)=−1*(4−49)−6*(7+4)

det(A)=45−66=−21

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

(C_11)=−42,(C_12)=−5,(C_13)=−7

(C_21)=24,(C_22)=−13,(C_23)=4

(C_31)=−45,(C_32)=−8,(C_33)=11

adj(A)=[[−42,24,−45],[−5,−13,−8],[−7,4,11]]

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

A(−1)=1/(−21)*[[−42,24,−45],[−5,−13,−8],[−7,4,11]]

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

X=1/(−21)*[[(−42)*(−4)+(24)*(−7)+(−45)*(0)],[(−5)*(−4)+(−13)*(−7)+(−8)*(0)],[(−7)*(−4)+(4)*(−7)+(11)*(0)]]

X=1/(−21)*[[168−168],[20+91],[28−28]]

X=1/(−21)*[[0],[111],[0]]

6. Simplify the results to find the values of x y and z

x=0

y=−111/21=−37/7

z=0

Final Answer

(x,y,z)=(0,−37/7,0)

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