If $\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix},$ then its transpose is given by
\[\mathbf{A}^T = \begin{pmatrix} a & c \\ b & d \end{pmatrix}.\]Given that $\mathbf{A}^T = \mathbf{A}^{-1},$ find $a^2 + b^2 + c^2 + d^2.$
From $\mathbf{A}^T = \mathbf{A}^{-1},$ $\mathbf{A}^T \mathbf{A} = \mathbf{I}.$  Hence,
\[\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} a & c \\ b & d \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}.\]Then $a^2 + b^2 = 1$ and $c^2 + d^2 = 1,$ so $a^2 + b^2 + c^2 + d^2 = \boxed{2}.$