Quantum computation is an interdisciplinary field that combines computer technology with quantum mechanics. It utilises quantum superposition, entanglement, and interference to revolutionise solving complex problems. Unlike classical computing, which relies on deterministic bit manipulation, quantum computing is based on qubit that represents two basis quantum state vectors of unit length in a complex Hilbert space. In quantum mechanics, we address various operators to represent observables for dynamical variables, but in quantum computation, we address quantum gates as operators to frame quantum circuits for carrying out various computational tasks. The essential components of quantum circuits are the quantum gates and they perform unitary transformations preserving probability amplitudes and thus enabling quantum parallelism. Operators in quantum mechanics are always represented in matrix as they connect two different states and thus in an analogy quantum gates in quantum computation frame represent as matrix. The goal of this paper is to explore individual quantum gates (single and multi qubit) and their series–parallel combinations through their matrix representation, to analyse their role in computing state evolution, commutation relation, probability measurement, and expectation values. Since quantum computation is based on linear algebra, characterisation of Hilbert space using the reversibility and linearity of quantum gates is carried out. In this research work matrix representation of rotational quantum gates and universal quantum gate are also addressed. The construction of a single qubit quantum gate using rotational gates is also addressed in this paper.