Find the Cofactor Matrix

Problem

[[1,0,−2,2],[−3,−1,0,−3],[−3,−2,2,3],[−3,1,0,0]]

Solution

1. Define the cofactor (C_i*j) of an element (a_i*j) in a matrix A as (C_i*j)=(−1)(i+j)*(M_i*j) where (M_i*j) is the determinant of the 3×3 submatrix formed by deleting the ith row and jth column.

2. Calculate the cofactors for the first row:

(C_11)=+det(−1)=1*(0−(−6))=6

(C_12)=−det(−3)=−(−1)*(−3*(0−0))=0

(C_13)=+det(−3)=−3*(−3−6)−1*(9−9)=27

(C_14)=−det(−3)=−(−2*(−3−3))=−12

3. Calculate the cofactors for the second row:

(C_21)=−det(0)=−1*(−6−4)=10

(C_22)=+det(1)=−3*(−6−4)=30

(C_23)=−det(1)=−(1*(0−3)+2*(−3−6))=21

(C_24)=+det(1)=1*(0−2)−2*(−3−6)=16

4. Calculate the cofactors for the third row:

(C_31)=+det(0)=1*(6−0)=6

(C_32)=−det(1)=−(−(−2)*(0−9))=18

(C_33)=+det(1)=1*(0−(−3))+2*(−3−3)=−9

(C_34)=−det(1)=−(−(−2)*(−3−3))=−12

5. Calculate the cofactors for the fourth row:

(C_41)=−det(0)=−(−(−2)*(−3−6)+2*(−2−0))=22

(C_42)=+det(1)=1*(0−(−6))−(−2)*(−9−9)+2*(−6−0)=−42

(C_43)=−det(1)=−(1*(−3−6)+2*(6−3))=3

(C_44)=+det(1)=1*(−2−0)−2*(6−3)=−8

6. Assemble the results into the cofactor matrix.

Final Answer

Cofactor Matrix=[[6,0,27,−12],[10,30,21,16],[6,18,−9,−12],[22,−42,3,−8]]

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