  Motivated by variational problems in nonlinear elasticity depending on the
deformation gradient and its inverse, we completely and explicitly describe
Young measures generated by matrix-valued mappings $\{Y_k\}_{k\in\N} \subset
L^p(\O;\R^{n\times n})$, $\O\subset\R^n$, such that $\{Y_k^{-1}\}_{k\in\N}
\subset L^p(\O;\R^{n\times n})$ is bounded, too. Moreover, the constraint $\det
Y_k>0$ can be easily included and is reflected in a condition on the support of
the measure. This condition typically occurs in problems of
nonlinear-elasticity theory for hyperelastic materials if $Y:=\nabla y$ for
$y\in W^{1,p}(\O;\R^n)$. Then we fully characterize the set of Young measures
generated by gradients of a uniformly bounded sequence in
$W^{1,\infty}(\O;\R^n)$ where the inverted gradients are also bounded in
$L^\infty(\O;\R^{n\times n})$. This extends the original results due to D.
Kinderlehrer and P. Pedregal.
