When a matrix is nonsingular, it means that it has an inverse. In other words, there exists another matrix that when multiplied by the original matrix, results in the identity matrix (a matrix with 1s on the main diagonal and 0s everywhere else).
The matrix C^(k+1) is the result of multiplying the matrix C by itself k+1 times. In other words, it is the k+1 power of the matrix C.
The matrix I-C is the result of subtracting the matrix C from the identity matrix I. In other words, it is the inverse of the matrix C.
This condition implies that the matrix C^(k+1) is equal to the zero matrix, meaning that all of its elements are 0. Since the zero matrix has no inverse, it follows that the matrix I-C, which is the inverse of C, must be nonsingular.
Yes, this proof can be applied to all square matrices as long as the condition C^(k+1)=0 is met.