Find the Inverse [[2,3,-4],[0,-4,2],[1,-1,5]]

Problem

[[2,3,−4],[0,−4,2],[1,−1,5]](−1)

Solution

1. Calculate the determinant of the matrix A using cofactor expansion along the first column.

det(A)=2*((−4)*(5)−(2)*(−1))−0+1*((3)*(2)−(−4)*(−4))

det(A)=2*(−20+2)+(6−16)

det(A)=2*(−18)−10=−36−10=−46

2. Find the matrix of minors by calculating the determinant of the 2×2 matrix remaining after removing the row and column of each element.

(M_11)=−18,(M_12)=−2,(M_13)=4

(M_21)=11,(M_22)=14,(M_23)=−5

(M_31)=−10,(M_32)=4,(M_33)=−8

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

C=[[−18,2,4],[−11,14,5],[−10,−4,−8]]

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

adj(A)=[[−18,−11,−10],[2,14,−4],[4,5,−8]]

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

A(−1)=1/(−46)*[[−18,−11,−10],[2,14,−4],[4,5,−8]]

Final Answer

[[2,3,−4],[0,−4,2],[1,−1,5]](−1)=[[9/23,11/46,5/23],[−1/23,−7/23,2/23],[−2/23,−5/46,4/23]]

Want more problems? Check here!