Find the cofactor matrix of the given 3x3 matrix A = [[1, 2, 3], [0, 4, 5], [1, 0, 1]]

Step 1 ：Step 1: Determine the minor matrix of A. The minor of an element in a matrix is the determinant of the sub-matrix formed by deleting the row and column in which the element lies. For 3x3 matrix, minor of a11 will be determinant of 2x2 matrix formed by deleting first row and first column. Similarly, we can find minors of other elements.

Step 2 ：Minor matrix, M = \[\begin{bmatrix} \det\begin{bmatrix}4 & 5\\0 & 1\end{bmatrix} & \det\begin{bmatrix}0 & 5\\1 & 1\end{bmatrix} & \det\begin{bmatrix}0 & 4\\1 & 0\end{bmatrix}\\ \det\begin{bmatrix}2 & 3\\0 & 1\end{bmatrix} & \det\begin{bmatrix}1 & 3\\1 & 1\end{bmatrix} & \det\begin{bmatrix}1 & 2\\1 & 0\end{bmatrix}\\ \det\begin{bmatrix}2 & 3\\4 & 5\end{bmatrix} & \det\begin{bmatrix}1 & 3\\0 & 5\end{bmatrix} & \det\begin{bmatrix}1 & 2\\0 & 4\end{bmatrix}\end{bmatrix}\]

Step 3 ：Step 2: Apply the rule of signs to the minor matrix to get the cofactor matrix. Recall that the rule of signs looks like a checkerboard of positive and negative signs across the matrix.

Step 4 ：Cofactor matrix, C = \[\begin{bmatrix} \det\begin{bmatrix}4 & 5\\0 & 1\end{bmatrix} & -\det\begin{bmatrix}0 & 5\\1 & 1\end{bmatrix} & \det\begin{bmatrix}0 & 4\\1 & 0\end{bmatrix}\\ -\det\begin{bmatrix}2 & 3\\0 & 1\end{bmatrix} & \det\begin{bmatrix}1 & 3\\1 & 1\end{bmatrix} & -\det\begin{bmatrix}1 & 2\\1 & 0\end{bmatrix}\\ \det\begin{bmatrix}2 & 3\\4 & 5\end{bmatrix} & -\det\begin{bmatrix}1 & 3\\0 & 5\end{bmatrix} & \det\begin{bmatrix}1 & 2\\0 & 4\end{bmatrix}\end{bmatrix}\]