How to tell if a random 3x3 Matrix is invertible

I have to produce a random 3x3 matrix A that is invertible and display it. I have a couple questions:

1. How do I know when a matrix is invertible? I used the command (inv) on the random 3x3 matrix that I had created and I got a 3x3 matrix with different numbers. Does this mean that the matrix is invertible?
2. I also got a hint with the question: Use a while-loop until you get one with non-zero determinant. I am confused by this because I used the determinants command (det) on my 3x3 matrix and got a nonzero determinant. I feel like I might be missing something here.

[Walter Roberson]

rank()

Most random matrices with floating point entries are invertible. Not all, but most. So unless you are using integer random values, do not be surprised if the first one you generate works.

[Stephen23]

https://www.mathworks.com/matlabcentral/answers/400327-why-is-det-a-bad-way-to-check-matrix-singularity

https://www.danielcolquitt.com/journal/2015/11/balancing-precision-speed

Answer 1

[Meg Noah]

Back to your question, I have to produce a random 3x3 matrix A that is invertible and display it. One way could be to start with a matrix that you know will have a determinant of zero and then add random noise to each element. It worked for me to generate random matrices that are invertable.

for MC = 1:10000
    % first create a matrix that you know has a low rcond value:
    A = double(uint32(1000.*rand(3,1)).*uint32(1000.*rand(1,3)));
    % then add noise
    C = A + 100.0*rand(3,3);
    if (rcond(C)<1e-20)
        disp('algorithm fails');
        C
        inv(C)
    end
end

There were objections to this suggestion about checking the determinant value. I had said: If the determinant of a square matrix is 0, it can't be inverted. I'd suggestion to test with - using your tolerance on the last argument. See comments below.

[John D'Errico]

Please don't tell people to use det to learn if a matrix is singular. That is just bad advice. Terrible advice even.

[Walter Roberson]

Looks like they are doing an assignment that recommended or requires det()

[John D'Errico]

I know. I just spent the last 40 minutes explaining the issues, and why you cannot just use det or inv to know if a matrix is singular, when my computer crashed mid-edit.

In fact, any simple tolerance using a tool like ismembertol or whatever will fail for det. And that is what I was going to say.

[Steven Lord]

If the determinant of a square matrix is 0, it can't be inverted.

This is a case where there's a difference between theory and practice. Here's a very simple example. determinantOfA is exactly 0 (not very small, but 0 to the last bit.) But A is just a non-zero scalar multiple of the identity matrix, and B is its inverse.

n = 1000;
A = (1/4)*eye(n);
determinantOfA = det(A) % exactly 0
B = inv(A); % but A has an inverse

isAtimesBIdentity = isequal(A*B, eye(n)) % true

The matrix B is quite simple. It only has n non-zero values, all of which happen to be exactly 4.

disp("B has " + numel(B) + " elements, " + ...
     nnz(B == 0) + " of which are 0 and " + ...
     nnz(B == 4) + " of which are 4.")

[Meg Noah]

There is a function in matlab:

C = cond(A)

that might be useful.

As bad as it is, I'd probably still do the terrible bad advice thing and go with something like:

if ((abs(det(A)) < 1e-27) && max(max(abs(A./A(1,1)-eye(size(A))))) ~= 0)
    disp(['it is NOT invertible ' num2str(rcond(A))]);
    rcond(A);
    B = inv(A); % but A has an inverse
else
    disp(['it IS invertible ' num2str(rcond(A))]);
    rcond(A);
    B = inv(A); % but A has an inverse
end

Putting it in a loop with A intialized by:

A = double(uint16(100.*rand(3,1)).*uint16(100.*rand(1,3)));

Showed that it can catch close to singular or badly scaled very well. It looks like that error is being thrown on the basis of the

rcond

routine from LAPACK. If rcond is small then the error is thrown. If rcond is close to 1 then it is considered an ok inverse. Another test could be the value of rcond(A) or cond(A).

[Steven Lord]

Better is simply to not invert a matrix. If you're trying to invert the matrix to solve a system of equations, use the backslash operator (\).

[Meg Noah]

But that wasn't the question. He has a task to produce a matrix that can be inverted.