Read Section 1.6. Further Results on Systems of Equations and
Invertibility.
For now you could skip the proof for Thm 1.6.1. We will do
it in class.
You'll notice that Thm 1.6.4 is essentially the long
theorem we're in the middle of proving.
Pay particular attention to Examples 1 and 2 concerning
invertible matrices (see Problems 1, 9a, 11). These
examples show, for a given invertible matrix A, how to use the
inverse matrix A-1 for solving various systems
Ax=b.
Pay particular attention to Examples 3,4 concerning
noninvertible matrices. They show how to determine
consistency by elimination (see problem 11). In other words,
they show, for a given singularmatrix A, how to
determine the particular column matrices b for which the
system Ax=b is consistent.
Read Section 1.7. Diagonal, Triangular, and Symmetric
Matrices.
We saw in class today one reason that diagonal matrices are
particularly nice. Triangular and symmetric matrices are also
particularly nice. This section explores some of the
reasons.
We probably won't spend any class time on 1.7, so make a
note of any questions you may have and either post on Q&A
or confer with others or confer with me to settle your
questions.