Module
Module 1: Matrices and Linear Algebra
Topics Covered
Matrix basics: trace, determinant, adjoint, inverse, and A(adj A) = |A|I
Rank of a matrix by minors, echelon form, and normal form
Consistency of AX = B by Rouche-Capelli theorem
Homogeneous and non-homogeneous linear systems
Eigenvalues, eigenvectors, and properties of symmetric matrices
Cayley-Hamilton theorem and inverse by CH
Diagonalization by P^-1 A P = D
Quadratic forms, canonical form, and nature test by eigenvalues
Positive definite, negative definite, indefinite, and semi-definite forms
Key Theorems and Formulae
tr(A) = sum of diagonal elements, adj(A) = transpose of cofactor matrix
A^-1 = adj(A) / |A|, so inverse exists only when |A| != 0
rho(A) = number of non-zero rows in echelon form
AX = B is consistent iff rho(A) = rho([A|B])
rho(A) = rho([A|B]) = n gives unique solution
sum of eigenvalues = tr(A), product of eigenvalues = |A|
Cayley-Hamilton: P(A) = 0, useful for A^-1 and powers of A
Diagonalization: P^-1 A P = D
Q(X) = X^T A X, nature depends on eigenvalue signs
Solved Example Types
Rank example: for [[1,2,3],[2,4,6],[3,6,9]], use R2 -> R2 - 2R1 and R3 -> R3 - 3R1, so only one non-zero row remains and rank = 1.
Consistency example: solve x + y + z = 6, 2x + 3y + z = 11, 3x + y + 2z = 10 by row reduction; rho(A) = rho([A|B]) = 3 gives unique solution x = 1/3, y = 7/3, z = 10/3.
Eigenvalue setup: form |A - lambda I| for [[2,-1,2],[-1,2,-1],[2,-1,2]] and solve for eigenvalues and eigenvectors.
Verify Cayley-Hamilton for [[1,2],[3,4]] and use it to find A^-1.
Reduce Q = 3x^2 + 3y^2 + 3z^2 - 2xy - 2yz - 2zx to canonical form and identify it as positive definite.
Exercise Sets
Basic: find rank of [[1,2,3],[2,3,4],[3,4,5]], solve 2x+y-z=3, x+2y+z=4, x+y+z=3, find eigenvalues of [[4,1],[2,3]], and verify A(adj A) = |A|I for a 2x2 matrix.
Intermediate: use Cayley-Hamilton to find A^-1 for [[2,1,1],[1,2,1],[1,1,2]], diagonalize [[6,-2,2],[-2,3,-1],[2,-1,3]], reduce x^2+y^2+z^2-2xy-2yz+2zx, and check consistency of x+2y+3z=1, 2x+3y+4z=2, 3x+4y+5z=3.
Advanced: if A^3 - 6A^2 + 11A - 6I = 0 find A^-1 and A^4, find index and signature for eigenvalues 2,-1,0, prove eigenvectors of distinct eigenvalues are linearly independent, and solve AX = B for singular A using generalized inverse ideas.
Book Mapping
Quick reference: rho(A) = non-zero rows in echelon form, AX = B consistent iff rho(A) = rho([A|B]), sum of eigenvalues = trace(A), product = |A|, diagonalizable iff independent eigenvectors exist, and A^-1 exists iff |A| != 0 and rho(A) = n. Book mapping: B.S. Grewal Ch.1, H.K. Das Ch.1, Kreyszig Ch.7.2-7.9