Let $G$ be a simple graph with vertex set $V(G) = {v_1, v_2,ldots, v_n}$ and $d_i$ the degree of its vertex $v_i$, $i = 1, 2, dots, n$. Inspired by the Randic matrix and the general Randic index of a graph, we introduce the concept of general Randi'c matrix $textbf{R}_alpha$ of $G$, which is defined by $(textbf{R}_alpha)_{i,j}=(d_id_j)^alpha$ if $v_i$ and $v_j$ are adjacent, and zero otherwise. Similarly, the general Randic eigenvalues are the eigenvalues of the general Randic} matrix, the greatest general Randic eigenvalue is the general Randic spectral radius of $G$, and the general Randic energy is the sum of the absolute values of the general Randic eigenvalues. In this paper, we prove some properties of the general Randi'c matrix and obtain lower and upper bounds for general Randic energy, also, we get some lower bounds for general Randic spectral radius of a connected graph. Moreover, we give a new sharp upper bound for the general Randic energy when $alpha=-1/2$.