  For given $\epsilon>0$ and $b\in\mathbb{R}^m$, we say that a real $m\times n$
matrix $A$ is $\epsilon$-badly approximable for the target $b$ if
$$\liminf_{q\in\mathbb{Z}^n, \|q\|\to\infty} \|q\|^n \langle Aq-b \rangle^m
\geq \epsilon,$$ where $\langle \cdot \rangle$ denotes the distance from the
nearest integral point. In this article, we obtain upper bounds for the
Hausdorff dimensions of the set of $\epsilon$-badly approximable matrices for
fixed target $b$ and the set of $\epsilon$-badly approximable targets for fixed
matrix $A$. Moreover, we give an equivalent Diophantine condition of $A$ for
which the set of $\epsilon$-badly approximable targets for fixed $A$ has full
Hausdorff dimension for some $\epsilon>0$. The upper bounds are established by
effectivizing entropy rigidity in homogeneous dynamics, which is of independent
interest. For the $A$-fixed case, our method also works for the weighted
setting where the supremum norms are replaced by certain weighted quasinorms.
