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Matrices Exercise (with Solution)
1. Basics of Matrices
2. Matrix Operations
3. Determinants Exercise
- Properties of Determinants Exercise
- Cofactor Expansion Exercise
- Determinant of a Triangular Matrix Exercise
- Applications of Determinants (Area, Volume, etc.) Exercise
4. Inverse and Rank
- Inverse of a Matrix (using Adjoint and Determinant)
- Properties of Inverse Matrices
- Rank of a Matrix
- Echelon Forms (Row and Reduced Row Echelon Form)
- Nullity and Column Space
5. Systems of Linear Equations
- Representation as a Matrix Equation
- Gauss Elimination Method
- Gauss-Jordan Method
- Cramer’s Rule
- Matrix Methods for Solutions
6. Special Matrix Forms
- Diagonalization
- Triangular Matrices (Upper and Lower)
- Block Matrices
- Sparse Matrices
- Vandermonde Matrix
- Toeplitz and Circulant Matrices
7. Eigenvalues and Eigenvectors
- Definition and Computation
- Properties of Eigenvalues and Eigenvectors
- Spectral Theorem
- Diagonalization using Eigenvalues
- Applications in Physics and Engineering
8. Advanced Matrix Topics
- Singular Value Decomposition (SVD)
- QR Decomposition
- LU Decomposition
- Cholesky Decomposition
- Moore-Penrose Pseudoinverse
9. Matrix Functions
- Exponential of a Matrix
- Logarithm of a Matrix
- Matrix Power
- Matrix Polynomial
10. Applications of Matrices
- Graph Theory (Adjacency and Incidence Matrices)
- Markov Chains and Transition Matrices
- Image and Signal Processing (Fourier Transforms)
- Linear Transformations in Geometry
- Data Science and Machine Learning (PCA, Regression)
11. Specialized Topics
- Tensor Algebra and Higher-Dimensional Arrays
- Non-Negative Matrices and Perron-Frobenius Theorem
- Matrix Factorization Techniques (e.g., NMF, SVD)
- Matrix Calculus
- Numerical Stability and Matrix Computation
- Applications in Cryptography (Hill Cipher)
