Matrix Equation Ax = b Flashcards

Matrix Equation Overview

• The equation Ax=b is classified as a matrix equation, with A being a matrix.
• A vector b is a linear combination of the columns of A if and only if the equation Ax=b has at least one solution.

Consistency of the Equation

• The equation Ax=b is not necessarily consistent if the augmented matrix [ A b ] has a pivot position in every row; one pivot could be in the column for b.
• If the columns of an mxn matrix A span ℝ^m, Ax=b is consistent for every b in ℝ^m, ensuring a solution exists.

Pivots and Solutions

• Inconsistent equations indicate that A cannot have a pivot position in every row, meaning no solution exists for some b.
• The first entry in the product Ax represents a sum of products, calculated from the corresponding entries in x and the first column of A.