Is it compulsory that when the determinant of a matrix is zero, then the matrix will have at least one equivalent matrix with a zero row or column?

Question

I think it is true because I can't find a matrix that isin't following the above statement. A little help would be appreciated!

Answer 1

Two matrices $A,B$ are equivalent, when there are two invertiable matrices $U,V$ so that $B=UAV$ holds. If we use a singular value decomposition $A = U \Sigma V^T$, you get $\Sigma$ as diagonal matrix with the singular values of $A$ on the diagonal. Since the determinant of $A$ is zero, there is at least one singular value $\sigma_n=0$. Since $U$ and $V$ are orthogonal matrices, they are invertible.

Short: $A$ is equivalent to $\Sigma$, the matrix of it's singular values. One of the singular values is zero, therefore a complete collumn and a complete row is zero.