In this paper, we first give new generalizations for third-order Horadam $\{H_{n}^{(3)}\}_{n\in \mathbb{N}}$ and generalized Tribonacci $\{h_{n}^{(3)}\}_{n\in \mathbb{N}}$ sequences for classic Horadam and generalized Fibonacci numbers. Considering these sequences, we define the matrix sequences which have elements of $\{H_{n}^{(3)}\}_{n\in \mathbb{N}}$ and $\{h_{n}^{(3)}\}_{n\in \mathbb{N}}$. Then we investigate their properties.
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