Certain Matrices Associated with Balancing and Lucas-balancing Numbers

Balancing numbers n and balancers r are originally defined as the solution of the Diophantine equation 1 + 2 + · · · + (n − 1) = (n + 1) + (n + 2) + · · · + (n + r). These numbers can be generated by the linear recurrence Bn+1 = 6Bn−Bn−1 or by the nonlinear recurrence B2 n = 1+Bn−1Bn+1. There is another way to generated balancing numbers using powers of a matrix QB = left(begin{array}-{cc} 6 -1 1 0 end{array} right). The matrix representation, indeed gives many known and new formulas for balancing numbers. In this paper, using matrix algebra we obtain several interesting results on balancing and related numbers.