Find the Inverse [[a,-2b,3d],[4a,b,-d],[2a,-b,3d]]

Problem

[[a,−2*b,3*d],[4*a,b,−d],[2*a,−b,3*d]]

Solution

1. Calculate the determinant of the matrix using the first row expansion.

det(A)=a*(3*b*d−b*d)−(−2*b)*(12*a*d−(−2*a*d))+3*d*(−4*a*b−2*a*b)

det(A)=a*(2*b*d)+2*b*(14*a*d)+3*d*(−6*a*b)

det(A)=2*a*b*d+28*a*b*d−18*a*b*d

det(A)=12*a*b*d

2. Find the matrix of cofactors by calculating the minor for each element and applying the sign pattern.

(C_11)=+(3*b*d−b*d)=2*b*d

(C_12)=−(12*a*d+2*a*d)=−14*a*d

(C_13)=+(−4*a*b−2*a*b)=−6*a*b

(C_21)=−(−6*b*d+3*b*d)=3*b*d

(C_22)=+(3*a*d−6*a*d)=−3*a*d

(C_23)=−(−a*b+4*a*b)=−3*a*b

(C_31)=+(2*b*d−3*b*d)=−b*d

(C_32)=−(−a*d−12*a*d)=13*a*d

(C_33)=+(a*b+8*a*b)=9*a*b

3. Form the adjugate matrix by taking the transpose of the cofactor matrix.

adj(A)=[[2*b*d,3*b*d,−b*d],[−14*a*d,−3*a*d,13*a*d],[−6*a*b,−3*a*b,9*a*b]]

4. Apply the inverse formula by dividing the adjugate matrix by the determinant 12*a*b*d

A(−1)=1/(12*a*b*d)*[[2*b*d,3*b*d,−b*d],[−14*a*d,−3*a*d,13*a*d],[−6*a*b,−3*a*b,9*a*b]]

5. Simplify each term by canceling common variables in the fractions.

(2*b*d)/(12*a*b*d)=1/(6*a)

(3*b*d)/(12*a*b*d)=1/(4*a)

(−b*d)/(12*a*b*d)=−1/(12*a)

(−14*a*d)/(12*a*b*d)=−7/(6*b)

(−3*a*d)/(12*a*b*d)=−1/(4*b)

(13*a*d)/(12*a*b*d)=13/(12*b)

(−6*a*b)/(12*a*b*d)=−1/(2*d)

(−3*a*b)/(12*a*b*d)=−1/(4*d)

(9*a*b)/(12*a*b*d)=3/(4*d)

Final Answer

[[a,−2*b,3*d],[4*a,b,−d],[2*a,−b,3*d]](−1)=[[1/(6*a),1/(4*a),−1/(12*a)],[−7/(6*b),−1/(4*b),13/(12*b)],[−1/(2*d),−1/(4*d),3/(4*d)]]

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