I have to determine whether a $4\times 4$ matrix $A$ is invertible. Suppose that there are no zero columns or zero rows. Is there any rule of thumb saying how many zero entries can be at most in $A$, or how many zero entries per column can be at most in $A$, or similar, in order to have $A$ invertible?
2026-04-02 13:01:54.1775134914
Rule of thumb on number of zero entries for invertibility of a $4\times 4 $ matrix?
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well, a diagonal matrix with nonzero entries along the main diagonal is always invertible, so to answer the original question there is no rule of thumb. Furthermore, a matrix that is all nonzero may be non-invertible. You would need to look at the rank of the matrix (or many, many other things, like the determinant)