Applications and Examples of Solving Linear Systems

Question 1

Define the rank of a matrix.

Answer 1

The rank of a matrix A – or rank(A) – is the number of leading ones (pivots) in any REF of A.

Question 2

What are two facts about the rank of a matrix?

Answer 2

1. The rank does NOT change when we do elementary row
   operations and rank(A) ≤ #rows or columns of A .

2. We can determine if a system is consistent or inconsistent by
comparing rank(A) and rank([A|b]).

Question 3

Suppose we row reduce [A|b] to RREF, what is the rank if all the leading ones are in the coefficient matrix of the
RREF?

Answer 3

rank(A) = rank([A|b])

Question 4

Suppose we row reduce [A|b] to RREF, what is the rank if there is a leading one in the column of constants of the
RREF?

Answer 4

rank(A) + 1 = rank([A|b])

Question 5

What is an implied by rank(A) = rank([A|b]) and rank(A) + 1 = rank([A|b])?

Answer 5

rank(A)≤rank([A|b])≤rank(A)+1

Question 6

If [A|b] is the augmented of a linear system, when is the system inconsistent?

Answer 6

When rank(A) < rank([A|b])

Question 7

If [A|b] is the augmented of a linear system, when does the system have a unique solution?

Answer 7

When rank(A) = rank([A|b]) = #cols of A

Question 8

If [A|b] is the augmented of a linear system, when does the system have infinitely many solution?

Answer 8

When rank(A) = rank([A|b]) < #cols of A