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If the determinant of an $n \times n$ matrix $A$ is zero then

1. rank of $A$ is $n$
2. rank of $A\leq n-2$
3. $A$ has at least one zero eigen value
4. the system of equations $Ax=0$ has no solution other than trivial solution

Product of Eigen values = Determinant
Determinant is 0 i.e. one of the Eigen Value is 0
for a singular matrix product of eigen value is zero

product of eigen value is equal det of matrix and sum is equal to trace