Inverses and the Transpose

Undoing a transformation, when that is possible.

12 problems · about 24 minutes · N-VM.C.10, A-REI.C.9

What this lesson teaches

The student computes 2 by 2 inverses and states when a matrix is invertible.

The inverse of A undoes A, and applying both in either order gives the identity.
For a 2 by 2, swap the diagonal, negate the off-diagonal, and divide by the determinant.
A determinant of zero means the plane was flattened, so no inverse can exist.

Straightforward practice. Get the method working first.

5 problems

Rows [5, 3] and [3, 2]. What is the determinant?
Answer 1
Why 1.
Why does a determinant of zero mean no inverse exists?
Answer The plane has been flattened onto a line, so the original cannot be recovered.
Why The collapse cannot be undone.
Rows [2, 0] and [0, 5]. What is the determinant?
Answer 10
Why 10 − 0.
Rows [1, 2] and [2, 4]. What is the determinant?
Answer 0
Why 4 − 4.
A times its inverse gives which matrix? Enter 1 for the identity, 2 for the zero matrix.
Answer 1
Why Nothing has moved.

The same ideas with more to keep track of.

3 problems

Rows [6, 2] and [1, 1]. What is the determinant?
Answer 4
Why 6 − 2.
Transposing a 2 by 5 matrix. How many rows does the result have?
Answer 5
Why Rows and columns swap.
A is invertible. How many solutions does Ax = b have?
Answer 1
Why Exactly one.

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

4 problems

Sort each matrix by whether it has an inverse.
Answer Invertible: Determinant 7, Determinant −3 · Singular: Determinant 0, Columns that are multiples of each other
Why Only a zero determinant blocks an inverse.
The identity matrix. What is its determinant?
Answer 1
Why Areas are unchanged.
The Determinant: A matrix with rows [3, 1] and [5, 2]. What is its determinant?
Answer 1
Why 1.
The Failure: A matrix with determinant 0. Does an inverse exist? 1 yes, 0 no.
Answer 0
Why No.