Circulant type matrices have become an important tool in solving fractional order differential equations. In this paper, we consider the circulant and left circulant and $g$-circulant matrices with the Jacobsthal and Jacobsthal-Lucas numbers. First, we discuss the invertibility of the circulant matrix and present the determinant and the inverse matrix. Furthermore, the invertibility of the left circulant and $g$-circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the left circulant and $g$-circulant matrices by utilizing the relation between left circulant, $g$-circulant matrices, and circulant matrix, respectively.