Credit card fraud remains a persistent and growing threat to financial institutions worldwide, leading to substantial financial losses and undermining consumer trust. This paper presents an advanced study on the application of supervised machine learning algorithms and linear algebra techniques for detecting fraudulent credit card transactions. Utilizing a comprehensive Credit Card Fraud Detection Dataset 2023 of over 550,000 anonymized credit card transactions made by European cardholders in 2023, we develop robust models capable of identifying fraudulent activities with high accuracy. Our approach leverages advanced machine learning algorithms, including Neural Networks and XGBoost classifiers, enhanced by linear algebra-based data preprocessing and feature engineering methods. The resulting models demonstrate significant improvements in predictive performance, offering a promising solution for real-time fraud detection in financial systems. This research not only advances the state of the art in fraud detection but also provides practical insights for financial institutions aiming to bolster their fraud prevention strategies.