The matrix $\left[\begin{array} &0 &1\\1& 0\end{array}\right]$
The following two solutions give different answers:
The first |A-λI|=0
(-λ) 1
1 (-λ)
= 0
∴(-λ)×(-λ)-1×1=0
∴(λ2)-1=0
∴(λ2-1)=0
∴(λ-1)(λ+1)=0
∴(λ-1)=0or(λ+1)=0
∴ The eigenvalues of the matrix A are given by λ=-1,1,
However, when I try to find the result in https://www.omnicalculator.com/math/singular-values it tell me the singular value is 1,1.
The reason is that "singular values" and "eigenvalues" are not the same thing.
You know what the eigenvalues of a matrix are. The singular values are the square roots of the eigenvalues of $A^\ast A$.
If $A$ is diagonal with non-negative eigenvalues, then these are the same thing, but in general, they are not the same (eg, singular values are always non-negative and real, but eigenvalues can be negative or complex)