We study the shifting property of a matrix and a sequence , i.e., the identity

when R is a Riordan matrix, a Sheffer matrix (exponential Riordan matrix), or a connection constants matrix (involving symmetric functions and continuants). Moreover, we consider the shifting identity for several sequences of combinatorial interest, such as the binomial coefficients, the polynomial coefficients, the Stirling numbers (and their q-analogues), the Lah numbers, the De Morgan numbers, the generalized Fibonacci numbers, the Bell numbers, the involutions numbers, the Chebyshev polynomials, the Stirling polynomials, the Hermite polynomials, the Gaussian coefficients, and the q-Fibonacci numbers.