I am the student of the pure mathematics and just want to share some tips about finding singularity or non-singularity of given system.
after converting the system into matrices form (just writing the coefficient in array) ,

if any two rows or any two columns are identical , the the whole system(matrix) is singular.

if any of the row or column is equal to zero (like all the element of any row or column is zero) ,
then the whole system (matrix) is singular

Very simple answer:
The determinant of the matrix usually shows the singularity. If the determinant of a matrix is zero, then the matrix is singular. Otherwise, it is non-singular. So, calculate the determinant of the matrix and check if it is zero or non-zero.
this will helpful.

finding determinent is also process for singularity but short process is to look the matrices and if you find two rows or colums equal or any of the row or colum is zero , then you dont need to find determinent for singlularity. you can directly say its singular matric

So can I say that the second tip is a subset of the rule, â€śif any row or column exists such that an integer k multiplied by some other row column gives that row or column, the system is singularâ€ť. In this case the k being 0?

To generalize the suggestion, we could say that a matrix is singular when at least one column (or row) of the matrix is a linear combination of other columns (or rows) of this matrix. In this case, we speak of collinear columns (or rows). For example if a1 = np.array([1, 2, 3]) and a2 = np.array([2,4,6]) are two columns (or rows) of the matrix, then a2 = 2*a1 . The two vectors are collinear, whose matrix is singular.

Good tip for small matrices that we can visually check, but can be computationally super expensive for larger matrices with tens of thousands of rows and columns (which is standard for deep learning). Thatâ€™s where determinants become useful.