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We introduce the so-called $d$-concavity, $d \geq 0,$ and prove that thenonsymmetric Monge-Amp\`{e}re type function of matrix variable is concave in anappropriate unbounded and convex set. We prove also the comparison principlefor nonsymmetric Monge-Amp\`{e}re type equations in the case when they areso-called $\delta$-elliptic with respect to compared functions with $0 \leq\delta <1$.

Elliptic solutions to nonsymmetric Monge-Amp\`{e}re type equations I. The $d$-concavity and the comparison principle