The Invertibility Theorem

Eight statements that are all the same statement.

12 problems · about 24 minutes · N-VM.C.10, A-REI.C.8

NameDate

Figure — use these to answer the problems

Every claim in this lesson is about this one picture. When the parallelogram survives, all eight statements hold; when it collapses, all eight fail.

Figure 1

Straightforward practice. Get the method working first.

5 problems

Rows [4, 8] and [1, 2]. Is the matrix invertible? 1 yes, 0 no.
Answer
2
Why do all these conditions agree?
- They all describe whether the transformation flattened space.
- It is a coincidence of the formulas.
Determinant 5. Is the matrix invertible? 1 yes, 0 no.
Answer
Determinant 0. Are the columns independent? 1 yes, 0 no.
Answer
An invertible 3 by 3 matrix. How many pivots?
Answer

The same ideas with more to keep track of.

3 problems

Ax = 0 has only the zero solution. Is A invertible? 1 yes, 0 no.
Answer
Rows [3, 6] and [1, 2]. Is the matrix invertible? 1 yes, 0 no.
Answer
The columns span the whole space. Is the determinant non-zero? 1 yes, 0 no.
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

Sort each statement by which side of the theorem it belongs to.
Write each item under the heading it belongs to: The determinant is non-zero · The columns are dependent · Every column has a pivot · Ax = 0 has a non-zero solution
True when A is invertible
True when A is singular
Rows [1, 0] and [0, 1]. Is the matrix invertible? 1 yes, 0 no.
Answer
The One Check
A square matrix has determinant 0. How many solutions does Ax = 0 have beyond the zero vector? Enter 0 for none, 1 for infinitely many.
Answer
The Rank
An invertible 4 by 4 matrix. How many pivots does it have?
Answer