The answer in Google search is,
If the determinant of a matrix is zero, then it has no inverse; hence the matrix is said to be singular. Only non-singular matrices have inverses.
Assume contrary to Google's answer that a singular matrix can have an inverse which is another singular matrix for example,
-1 2 0 2 2 0 -4 0
2 -1 2 0 0 2 0 -4
0 2 -1 2 -4 0 2 0
2 0 2 -1 0 -4 0 2
2 0 -4 0 -1 2 0
0 2 0 -4 2 -1 2 0
-4 0 2 0 0 2 -1 2
0 -4 0 2 2 0 2 -1
And,
11/105 22/105 8/105 22/105 22/105 8/105 4/105 8/105
22/105 11/105 22/105 8/105 8/105 22/105 8/105 4/105
8/105 22/105 11/105 22/105 4/105 8/105 22/105 8/105
22/105 8/105 22/105 11/105 8/105 4/105 8/105 22/105
22/105 8/105 4/105 8/105 11/105 22/105 8/105 22/105
8/105 22/105 8/105 4/105 22/105 11/105 22/105 8/105
4/105 8/105 22/105 8/105 8/105 22/105 11/105 22/105
8/105 4/105 8/105 22/105 22/105 8/105 22/105 11/105
So what?
Any matrix, square or strictly rectangular, always has a unique pseudo inverse ( The Moore Penrose Inverse).
For determinant to be defined, a matrix has to be square. A square singular matrix has a unique pseudo inverse with which it satisfy the Moore Penrose conditions 1,2,3 and 4.
For a square invertible matrix, it's inverse is unique and is equal to it's pseudo inverse.
First of all, in the fifth row there is a missing value, so that the matrix is not square.
Therefore, it is not clear what kind of calculation you have produced.
Consider the homogeneous system of n independent linear algebraic equations of the first order:
a11 x1 + . . +a1n xn =0
a21 x1+ . . +a2n xn=0
. . . .
an1x1+ . . +ann xn=0
In matrix format,
A . x = 0 . . . (2)
This system has a solution if and only if the matrix A nxn (ai,j) satisfies two apparently contradictory conditions,
i- The determinant of the matrix A =0.
ii- The inverse of A (A^-1) exists.
In this case, the solution for the vector x is given by,
x=A^-1 (b+S) . . . . (3)
b is the assumed Diriclet boundary condition vector and S is the source/sink term vector.
Equation 3 describes the entanglement in the control volume cube of n equidistant free nodes which have an important place in numerical statistical theory called Cairo techniques.
Therefore, if a matrix has zero determinant, can have an inverse then the average entanglement in quantum and classical physics also exist.
As an illustrative example, consider the singular matrix A,
0 1/6 0 1/6 1/6 0 0 0
1/6 0 1/6 0 0 1/6 0 0
0 1/6 0 1/6 0 0 1/6 0
1/6 0 1/6 0 0 0 0 1/6
1/6 0 0 0 0 1/6 0 1/6
0 1/6 0 0 1/6 0 1/6 0
0 0 1/6 0 0 1/6 0 1/6
0 0 0 1/6 1/6 0 1/6 0
iterative det [A]=0
But its inverse matrix exists as,
A^-1=
0 2 0 2 2 0 -4 0
2 0 2 0 0 2 0 -4
0 2 0 2 -4 0 2 0
2 0 2 0 0 -4 0 2
2 0 -4 0 0 2 0 2
0 2 0 -4 2 0 2 0
-4 0 2 0 0 2 0 2
0 -4 0 2 2 0 2 0
Note that A A^-1= I
Which means that the singular matrix A has a true (and not a pseudo) inverse
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