A Rank matrix is a square matrix that has non-zero determinant. The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. A matrix with a rank of n has n linearly independent rows and n linearly independent columns.
The rank of a matrix can be determined by performing row operations to reduce the matrix to its row echelon form. The number of non-zero rows in the reduced matrix is the rank of the original matrix.
A Co-factor matrix is a square matrix that represents the co-factors of the elements in a given matrix. The co-factor of an element in a matrix is the determinant of the sub-matrix obtained by deleting the row and column that contain the element.
The co-factor matrix can be calculated by finding the co-factor of each element in the given matrix and arranging them in a new matrix according to their positions. The resulting matrix is the co-factor matrix of the original matrix.
The rank of a matrix is equal to the number of non-zero elements in its co-factor matrix. If a matrix has a rank of n, its co-factor matrix will also have a rank of n. This means that the co-factor matrix can be used to determine the rank of a matrix.