How to tell if a random 3x3 Matrix is invertible

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Kyle Donk 2020-1-10

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编辑： Meg Noah 2020-1-11

采纳的回答： Meg Noah

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

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?
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.

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Walter Roberson 2020-1-10

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 2020-1-10

编辑：Stephen23 2020-1-10

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

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Meg Noah 2020-1-10

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编辑：Meg Noah 2020-1-11

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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.