1. The problem asks why minus signs appear outside the matrix elements when calculating the adjoint (adjugate) of matrix $A$. 2. The adjoint matrix is the transpose of the cofactor matrix. Each cofactor is calculated as $C_{ij} = (-1)^{i+j} M_{ij}$, where $M_{ij}$ is the minor determinant. 3. The factor $(-1)^{i+j}$ introduces a sign pattern (plus or minus) depending on the position $(i,j)$ in the matrix. 4. When calculating cofactors, the minus signs are placed outside the minor determinants to reflect this alternating sign pattern. 5. This is why you see minus signs outside the matrices or determinants in the adjoint calculation: they represent the $(-1)^{i+j}$ factor applied to each minor. 6. In summary, the minus signs are not arbitrary; they come from the definition of cofactors which are essential for finding the adjoint matrix. 7. This sign pattern ensures that the adjoint matrix correctly relates to the inverse matrix by $A^{-1} = \frac{1}{|A|} \mathrm{adj}(A)$. Final answer: Minus signs outside the matrices in the adjoint calculation represent the $(-1)^{i+j}$ factor in cofactors, which is necessary for correctly computing the adjoint matrix.