Find the Cofactor Matrix [[1,2,3],[0,4,5],[1,0,6]]

Problem

[[1,2,3],[0,4,5],[1,0,6]]

Solution

1. Identify the matrix A and the formula for the cofactor (C_i*j) of an element (a_i*j) which is (C_i*j)=(−1)(i+j)*(M_i*j) where (M_i*j) is the determinant of the 2×2 submatrix formed by deleting the ith row and jth column.

2. Calculate the cofactors for the first row:

(C_11)=(−1)(1+1)*det(4)=1*(24−0)=24

(C_12)=(−1)(1+2)*det(0)=−1*(0−5)=5

(C_13)=(−1)(1+3)*det(0)=1*(0−4)=−4

3. Calculate the cofactors for the second row:

(C_21)=(−1)(2+1)*det(2)=−1*(12−0)=−12

(C_22)=(−1)(2+2)*det(1)=1*(6−3)=3

(C_23)=(−1)(2+3)*det(1)=−1*(0−2)=2

4. Calculate the cofactors for the third row:

(C_31)=(−1)(3+1)*det(2)=1*(10−12)=−2

(C_32)=(−1)(3+2)*det(1)=−1*(5−0)=−5

(C_33)=(−1)(3+3)*det(1)=1*(4−0)=4

5. Assemble the cofactor matrix C using the calculated values (C_i*j)

Final Answer

Cofactor Matrix=[[24,5,−4],[−12,3,2],[−2,−5,4]]

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