1. Matrix Dimensions and Operations
Matrix: An m x n array. a_{ij} = entry in row i, column j.
| Operation | Rule | Requirement |
|---|
| Addition A + B | Add entry by entry | Same dimensions |
| Scalar cA | Multiply every entry by c | Any matrix |
| Product AB | Row-column dot product | A cols = B rows |
(m x n)(n x p) = m x p
2. Properties
Matrix multiplication: associative, distributive, NOT commutative. AB != BA in general.
AB = 0 does NOT imply A = 0 or B = 0. AB = AC does NOT imply B = C.
3. Transpose
A^T: Flip rows and columns. (A^T)_{ij} = a_{ji}. Size m x n becomes n x m.
(AB)^T = B^T A^T (order reverses). Symmetric: A^T = A. A^T A is always symmetric.
4. Inverse Matrices
A^{-1}: AA^{-1} = A^{-1}A = I. Only square matrices can be invertible. Singular = not invertible (det = 0).
2x2: A = [a b; c d], A^{-1} = (1/(ad-bc)) [d -b; -c a]
General method: Row reduce [A | I] to [I | A^{-1}].
(AB)^{-1} = B^{-1}A^{-1}. (A^T)^{-1} = (A^{-1})^T. Solving: x = A^{-1}b.
5. Elementary Matrices
Result of one row operation on I. EA performs that operation on A. Every elementary matrix is invertible.
6. LU Factorization
A = LU (L = lower triangular with multipliers, U = upper triangular REF)
To solve Ax = b: (1) Ly = b (forward sub), (2) Ux = y (back sub). Efficient for multiple right-hand sides.