Alzheimer's disease (AD) is a common neurodegenerative disease with an often seen prodromal mild cognitive impairment (MCI) phase, where memory loss is the main complaint progressively worsening with behavior issues and poor self-care. However, not all patients clinically diagnosed with MCI progress to the AD. Currently, several high-dimensional classification techniques have been developed to automatically distinguish among AD, MCI, and healthy control (HC) patients based on T1-weighted MRI. However, these method features are based on wavelets, contourlets, gray-level co-occurrence matrix, etc., rather than using clinical features which helps doctors to understand the pathological mechanism of the AD. In this study, a new approach is proposed using cortical thickness and subcortical volume for distinguishing binary and tertiary classification of the National Research Center for Dementia dataset (NRCD), which consists of 326 subjects. Five classification experiments are performed: binary classification, i.e., AD vs HC, HC vs mAD (MCI due to the AD), and mAD vs aAD (asymptomatic AD), and tertiary classification, i.e., AD vs HC vs mAD and AD vs HC vs aAD using cortical and subcortical features. Datasets were divided in a 70/30 ratio, and later, 70% were used for training and the remaining 30% were used to get an unbiased estimation performance of the suggested methods. For dimensionality reduction purpose, principal component analysis (PCA) was used. After that, the output of PCA was passed to various types of classifiers, namely, softmax, support vector machine (SVM), k-nearest neighbors, and naïve Bayes (NB) to check the performance of the model. Experiments on the NRCD dataset demonstrated that the softmax classifier is best suited for the AD vs HC classification with an F1 score of 99.06, whereas for other groups, the SVM classifier is best suited for the HC vs mAD, mAD vs aAD, and AD vs HC vs mAD classifications with the F1 scores being 99.51, 97.5, and 99.99, respectively. In addition, for the AD vs HC vs aAD classification, NB performed well with an F1 score of 95.88. In addition, to check our proposed model efficiency, we have also used the OASIS dataset for comparing with 9 state-of-the-art methods.