Find the Variables

Problem

[[1,1,0],[1,4,3],[2,4,1]]*X=[[2,−3,−2],[0,2,0],[4,1,1]]

Solution

1. Identify the matrix equation as A*X=B where A is the coefficient matrix and B is the result matrix. To find X calculate X=A(−1)*B

2. Calculate the determinant of A using the first row to ensure it is invertible.

det(A)=1*(4*(1)−3*(4))−1*(1*(1)−3*(2))+0

det(A)=1*(4−12)−1*(1−6)

det(A)=−8+5=−3

3. Find the matrix of cofactors for A

(C_11)=+(4−12)=−8

(C_12)=−(1−6)=5

(C_13)=+(4−8)=−4

(C_21)=−(1−0)=−1

(C_22)=+(1−0)=1

(C_23)=−(4−2)=−2

(C_31)=+(3−0)=3

(C_32)=−(3−0)=−3

(C_33)=+(4−1)=3

4. Form the adjugate matrix by transposing the cofactor matrix.

adj(A)=[[−8,−1,3],[5,1,−3],[−4,−2,3]]

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

A(−1)=−1/3*[[−8,−1,3],[5,1,−3],[−4,−2,3]]=[[8/3,1/3,−1],[−5/3,−1/3,1],[4/3,2/3,−1]]

6. Multiply A(−1) by B to find X

X=[[8/3,1/3,−1],[−5/3,−1/3,1],[4/3,2/3,−1]]*[[2,−3,−2],[0,2,0],[4,1,1]]

7. Compute the elements of the resulting matrix X

(X_11)=16/3+0−4=4/3

(X_12)=−8+2/3−1=−25/3

(X_13)=−16/3+0−1=−19/3

(X_21)=−10/3+0+4=2/3

(X_22)=5−2/3+1=16/3

(X_23)=10/3+0+1=13/3

(X_31)=8/3+0−4=−4/3

(X_32)=−4+4/3−1=−11/3

(X_33)=−8/3+0−1=−11/3

Final Answer

[[1,1,0],[1,4,3],[2,4,1]]*X=[[2,−3,−2],[0,2,0],[4,1,1]]⇒X=[[4/3,−25/3,−19/3],[2/3,16/3,13/3],[−4/3,−11/3,−11/3]]

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