  We introduce \underline{F}actor-\underline{A}ugmented \underline{Ma}trix
\underline{R}egression (FAMAR) to address the growing applications of
matrix-variate data and their associated challenges, particularly with
high-dimensionality and covariate correlations. FAMAR encompasses two key
algorithms. The first is a novel non-iterative approach that efficiently
estimates the factors and loadings of the matrix factor model, utilizing
techniques of pre-training, diverse projection, and block-wise averaging. The
second algorithm offers an accelerated solution for penalized matrix factor
regression. Both algorithms are supported by established statistical and
numerical convergence properties. Empirical evaluations, conducted on synthetic
and real economics datasets, demonstrate FAMAR's superiority in terms of
accuracy, interpretability, and computational speed. Our application to
economic data showcases how matrix factors can be incorporated to predict the
GDPs of the countries of interest, and the influence of these factors on the
GDPs.
