Rotating and non-linear magnetic-charged black hole with an anisotropic matter field

We present the solution of a non-linear magnetic-charged black hole with an anisotropic matter field and further extend it to obtain the corresponding rotating black hole solution using the modified Newman-Janis algorithm. The event horizon and ergosphere of the rotating black hole are studied in terms of the perspective of geometric properties, revealing that the rotating black hole can have up to three horizons. The first law of thermodynamics and the squared-mass formula for the rotating black hole are derived from a thermodynamic perspective, based on which we obtain the thermodynamic quantities and study the thermodynamic stability of the rotating black hole. Additionally, we calculate the Penrose process for the rotating black hole, indicating the influence of various black hole parameters on the maximal efficiency of the Penrose process.