Math 251 - Alexiades Review on Chapter 1
know and understand: concepts / definitions / main results Concepts

matrix operations, elementary row operations, REF, RREF, rank

invertible matrix; Fundamental Thm for Square Matrices

transpose, diagonal, triangular, symmetric matrices
Precise definition of

equal matrices, row-equivalent matrices, elementary matrix, A^T

invertible n×n matrix A   ( ∃ n×n matrix A^-1 such that A^-1A = AA^-1 = I )

symmetric matrix   ( A^T=A )
Methods: Know How To

write a linear m×n system in matrix form: Ax=b

Gauss elimination: reduce m×n matrix A to Row Echelon Form; find rank

Gauss-Jordan elimination: reduce m×n A to Reduced Row Echelon Form; find rank

decide consistency of m×n system Ax = b ; solve Ax = b

find A^-1 by Gauss-Jordan elimination (on [A | I ])
Basic / crucial Theorems to know

1.6.1 , 1.6.2 , 1.6.3 , 1.6.5(AB invertible iff A and B invertible)

Fundamental Theorem for square matrices:   (6 equivalent statements so far)
        For a square n×n matrix A, the following are equivalent:
      1. A is invertible
      2. A ∼ I_n   (i.e. A is row equivalent to the Identity)
      3. A is expressible as a product of elementary matrices
      4. rank(A) = n
      5. the homogeneous system Ax = 0 has only the trivial solution
      6. the nonhomogeneous system Ax = b has unique solution for any vector b ∈ Rⁿ

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Last Updated:   27 Aug 2011