Find the Adjoint [[1,0,0],[0,2,6],[0,-4,-12]]

Problem

adj*[[1,0,0],[0,2,6],[0,−4,−12]]

Solution

1. Identify the matrix A and the formula for the adjoint. The adjoint (or adjugate) is the transpose of the cofactor matrix C

A=[[1,0,0],[0,2,6],[0,−4,−12]]

2. Calculate the cofactor (C_11) by finding the determinant of the 2×2 matrix remaining after removing row 1 and column 1.

(C_11)=(−1)(1+1)*|[2,6],[−4,−12]|=1*((2)*(−12)−(6)*(−4))=0

3. Calculate the remaining cofactors for the first row.

(C_12)=(−1)(1+2)*|[0,6],[0,−12]|=−1*(0−0)=0

(C_13)=(−1)(1+3)*|[0,2],[0,−4]|=1*(0−0)=0

4. Calculate the cofactors for the second row.

(C_21)=(−1)(2+1)*|[0,0],[−4,−12]|=−1*(0−0)=0

(C_22)=(−1)(2+2)*|[1,0],[0,−12]|=1*(−12−0)=−12

(C_23)=(−1)(2+3)*|[1,0],[0,−4]|=−1*(−4−0)=4

5. Calculate the cofactors for the third row.

(C_31)=(−1)(3+1)*|[0,0],[2,6]|=1*(0−0)=0

(C_32)=(−1)(3+2)*|[1,0],[0,6]|=−1*(6−0)=−6

(C_33)=(−1)(3+3)*|[1,0],[0,2]|=1*(2−0)=2

6. Construct the cofactor matrix C

C=[[0,0,0],[0,−12,4],[0,−6,2]]

7. Transpose the cofactor matrix to find the adjoint.

adj(A)=CT=[[0,0,0],[0,−12,−6],[0,4,2]]

Final Answer

adj*[[1,0,0],[0,2,6],[0,−4,−12]]=[[0,0,0],[0,−12,−6],[0,4,2]]

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