1. The problem is to calculate the adjugate (or adjoint) matrix of a given square matrix. 2. The adjugate matrix is the transpose of the cofactor matrix. 3. Steps to find the adjugate matrix for a matrix $A$: - Find the cofactor of each element in $A$. The cofactor $C_{ij}$ is given by $$C_{ij} = (-1)^{i+j} M_{ij}$$ where $M_{ij}$ is the determinant of the submatrix formed by deleting the $i$-th row and $j$-th column. - Form the cofactor matrix by placing each cofactor $C_{ij}$ in position $(i,j)$. - Transpose the cofactor matrix to get the adjugate matrix: $$\text{adj}(A) = (C_{ij})^T$$ 4. Example: For a $2 \times 2$ matrix $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ - Cofactors: $$C_{11} = d, \quad C_{12} = -c, \quad C_{21} = -b, \quad C_{22} = a$$ - Cofactor matrix: $$\begin{bmatrix} d & -c \\ -b & a \end{bmatrix}$$ - Adjugate matrix (transpose): $$\text{adj}(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$ 5. This method generalizes to any $n \times n$ matrix by computing all cofactors and transposing the resulting matrix. The adjugate matrix is useful in finding the inverse of a matrix since $$A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$$ if $\det(A) \neq 0$.