Find the Inverse [[-7,-8,3],[6,4,4],[2,8,4]]

Problem

[[−7,−8,3],[6,4,4],[2,8,4]]

Solution

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

det(A)=−7*(4⋅4−4⋅8)−(−8)*(6⋅4−4⋅2)+3*(6⋅8−4⋅2)

det(A)=−7*(16−32)+8*(24−8)+3*(48−8)

det(A)=−7*(−16)+8*(16)+3*(40)

det(A)=112+128+120=360

2. Find the matrix of minors by calculating the determinant of the 2×2 matrix remaining when the row and column of each element are removed.

(M_11)=16−32=−16

(M_12)=24−8=16

(M_13)=48−8=40

(M_21)=−32−24=−56

(M_22)=−28−6=−34

(M_23)=−56−(−16)=−40

(M_31)=−32−12=−44

(M_32)=−28−18=−46

(M_33)=−28−(−48)=20

3. Apply the cofactor signs using the pattern +−+ −+− +−+ to create the cofactor matrix C

C=[[−16,−16,40],[56,−34,40],[−44,46,20]]

4. Transpose the cofactor matrix to find the adjugate matrix adj(A)

adj(A)=[[−16,56,−44],[−16,−34,46],[40,40,20]]

5. Divide by the determinant to find the inverse matrix A(−1)=1/det(A)*adj(A)

A(−1)=1/360*[[−16,56,−44],[−16,−34,46],[40,40,20]]

6. Simplify the fractions by dividing each element by 360

A(−1)=[[−2/45,7/45,−11/90],[−2/45,−17/180,23/180],[1/9,1/9,1/18]]

Final Answer

[[−7,−8,3],[6,4,4],[2,8,4]](−1)=[[−2/45,7/45,−11/90],[−2/45,−17/180,23/180],[1/9,1/9,1/18]]

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