The Invertibility Theorem

Eight statements that are all the same statement.

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

What this lesson teaches

The student connects invertibility, determinants, independence, rank and solution counts.

For a square matrix, invertibility, a non-zero determinant, independent columns and unique solutions are all one fact.
If one condition fails, every one of them fails.
They agree because they all describe whether the transformation flattened space.

Straightforward practice. Get the method working first.

5 problems

Rows [4, 8] and [1, 2]. Is the matrix invertible? 1 yes, 0 no.
Answer 0
Why No.
Why do all these conditions agree?
Answer They all describe whether the transformation flattened space.
Why One geometric event, many descriptions.
Determinant 5. Is the matrix invertible? 1 yes, 0 no.
Answer 1
Why Non-zero determinant.
Determinant 0. Are the columns independent? 1 yes, 0 no.
Answer 0
Why All the conditions fail together.
An invertible 3 by 3 matrix. How many pivots?
Answer 3
Why One per column.

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 1
Why That is one of the equivalent conditions.
Rows [3, 6] and [1, 2]. Is the matrix invertible? 1 yes, 0 no.
Answer 0
Why 6 − 6 = 0.
The columns span the whole space. Is the determinant non-zero? 1 yes, 0 no.
Answer 1
Why Spanning and invertibility travel together.

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.
Answer True when A is invertible: The determinant is non-zero, Every column has a pivot · True when A is singular: The columns are dependent, Ax = 0 has a non-zero solution
Why Independence and invertibility go together.
Rows [1, 0] and [0, 1]. Is the matrix invertible? 1 yes, 0 no.
Answer 1
Why The identity.
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 1
Why Infinitely many.
The Rank: An invertible 4 by 4 matrix. How many pivots does it have?
Answer 4
Why 4.