Let $M_n$ be the algebra of all complex $n{ imes}n$ matrices and ${phi}:M_n{ ightarrow}M_n$ a surjective map (not necessarily additive or multiplicative) satisfying one of the following equations: $${det}({phi}(A){phi}(B)+{phi}(X))={det}(AB+X),;A,B,X{in}M_n,\{sigma}({phi}(A){phi}(B)+{phi}(X))={sigma}(AB+X),;A,B,X{in}M_n$$. Then it is an automorphism, where ${sigma}(A)$ is the spectrum of $A{in}M_n$. We also show that if $mathfrak{A}$ be a standard operator algebra, $mathfrak{B}$ is a unital Banach algebra with trivial center and if ${phi}:mathfrak{A}{ ightarrow}mathfrak{B}$ is a multiplicative surjection preserving spectrum, then ${phi}$ is an algebra isomorphism.