Chapter 1.6

Question 0

Equivalent statements (2) (5)

If A is an invertible square matrix then

Answer 0

1) Ax=0 has only the trivial solution
2) The reduced row echelon form of A is the identity matrix.
3) A is expressible as a product of elementary matrices
4) Ax=b is consistent for every nx1 matrix n
5) Ax=b has exactly 1 solution for every nx1 matrix b.

Question 1

If A is an invertible matrix of order n, what can be said about the system of equations Ax=b

Answer 1

For each nx1 matrix b, the system has exactly 1 solution, namely x=inv(A)b

Question 2

Let A and B be square matrices of the same size, if AB is invertible, then

Answer 2

A and B must also be invertible

Question 3

What is the fundamental problem

Answer 3

Let A be a fixed Mxn matrix, Find all mx1 matrices b such that the system Ax=b is consistent