Find the Adjoint [[3,4,-1],[2,-2,3],[1,-1,x]]

Problem

adj*[[3,4,−1],[2,−2,3],[1,−1,x]]

Solution

1. Identify the matrix A and the formula for the adjoint, which is the transpose of the cofactor matrix C

A=[[3,4,−1],[2,−2,3],[1,−1,x]]

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

(C_11)=(−1)(1+1)*det(−2)=−2*x+3

3. Calculate the cofactor (C_12) by removing row 1 and column 2.

(C_12)=(−1)(1+2)*det(2)=−(2*x−3)=−2*x+3

4. Calculate the cofactor (C_13) by removing row 1 and column 3.

(C_13)=(−1)(1+3)*det(2)=−2−(−2)=0

5. Calculate the cofactor (C_21) by removing row 2 and column 1.

(C_21)=(−1)(2+1)*det(4)=−(4*x−1)=−4*x+1

6. Calculate the cofactor (C_22) by removing row 2 and column 2.

(C_22)=(−1)(2+2)*det(3)=3*x+1

7. Calculate the cofactor (C_23) by removing row 2 and column 3.

(C_23)=(−1)(2+3)*det(3)=−(−3−4)=7

8. Calculate the cofactor (C_31) by removing row 3 and column 1.

(C_31)=(−1)(3+1)*det(4)=12−2=10

9. Calculate the cofactor (C_32) by removing row 3 and column 2.

(C_32)=(−1)(3+2)*det(3)=−(9−(−2))=−11

10. Calculate the cofactor (C_33) by removing row 3 and column 3.

(C_33)=(−1)(3+3)*det(3)=−6−8=−14

11. Construct the cofactor matrix C using the calculated values.

C=[[−2*x+3,−2*x+3,0],[−4*x+1,3*x+1,7],[10,−11,−14]]

12. Transpose the cofactor matrix to find the adjoint matrix.

adj(A)=CT

Final Answer

adj*[[3,4,−1],[2,−2,3],[1,−1,x]]=[[−2*x+3,−4*x+1,10],[−2*x+3,3*x+1,−11],[0,7,−14]]

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