1. The problem is to find the cofactor matrix of a given matrix. Since the matrix is not provided, let's assume a general 3x3 matrix: $$A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$$ 2. The cofactor matrix is composed of cofactors $C_{ij}$, where each cofactor is calculated as: $$C_{ij} = (-1)^{i+j} M_{ij}$$ Here, $M_{ij}$ is the minor of element $a_{ij}$, which is the determinant of the matrix formed by deleting the $i$-th row and $j$-th column. 3. Calculate each minor: - $M_{11} = \begin{vmatrix} e & f \\ h & i \end{vmatrix} = ei - fh$ - $M_{12} = \begin{vmatrix} d & f \\ g & i \end{vmatrix} = di - fg$ - $M_{13} = \begin{vmatrix} d & e \\ g & h \end{vmatrix} = dh - eg$ - $M_{21} = \begin{vmatrix} b & c \\ h & i \end{vmatrix} = bi - ch$ - $M_{22} = \begin{vmatrix} a & c \\ g & i \end{vmatrix} = ai - cg$ - $M_{23} = \begin{vmatrix} a & b \\ g & h \end{vmatrix} = ah - bg$ - $M_{31} = \begin{vmatrix} b & c \\ e & f \end{vmatrix} = bf - ce$ - $M_{32} = \begin{vmatrix} a & c \\ d & f \end{vmatrix} = af - cd$ - $M_{33} = \begin{vmatrix} a & b \\ d & e \end{vmatrix} = ae - bd$ 4. Apply the sign pattern $(-1)^{i+j}$ to each minor to get cofactors: $$C = \begin{bmatrix} +M_{11} & -M_{12} & +M_{13} \\ -M_{21} & +M_{22} & -M_{23} \\ +M_{31} & -M_{32} & +M_{33} \end{bmatrix}$$ 5. Substitute the minors: $$C = \begin{bmatrix} ei - fh & -(di - fg) & dh - eg \\ -(bi - ch) & ai - cg & -(ah - bg) \\ bf - ce & -(af - cd) & ae - bd \end{bmatrix}$$ This matrix $C$ is the cofactor matrix of $A$. Final answer: $$\text{Cofactor matrix} = \begin{bmatrix} ei - fh & fg - di & dh - eg \\ ch - bi & ai - cg & bg - ah \\ bf - ce & cd - af & ae - bd \end{bmatrix}$$