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Cogito

Linear Algebra · Chapter 3 · Lesson 3

Practice Worksheet

Inverses and the Transpose

Undoing a transformation, when that is possible.

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

NameDate

Figure — use these to answer the problems

Drive the determinant to zero and the parallelogram collapses. Information has been destroyed and no inverse can bring it back.
ij
  • i-hat lands on(3, 2)
  • j-hat lands on(1, 2)
  • Determinant4
a3
b1
c2
d2
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. Rows [5, 3] and [3, 2]. What is the determinant?

    Answer

  2. Why does a determinant of zero mean no inverse exists?

    • The plane has been flattened onto a line, so the original cannot be recovered.
    • The numbers become too large to compute.

  3. Rows [2, 0] and [0, 5]. What is the determinant?

    Answer

  4. Rows [1, 2] and [2, 4]. What is the determinant?

    Answer

  5. A times its inverse gives which matrix? Enter 1 for the identity, 2 for the zero matrix.

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. Rows [6, 2] and [1, 1]. What is the determinant?

    Answer

  2. Transposing a 2 by 5 matrix. How many rows does the result have?

    Answer

  3. A is invertible. How many solutions does Ax = b have?

    Answer

Stretch Yourself

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

4 problems

  1. Sort each matrix by whether it has an inverse.

    Write each item under the heading it belongs to: Determinant 7 · Determinant 0 · Determinant −3 · Columns that are multiples of each other

    Invertible

    Singular

  2. The identity matrix. What is its determinant?

    Answer

  3. The Determinant

    A matrix with rows [3, 1] and [5, 2]. What is its determinant?

    Answer

  4. The Failure

    A matrix with determinant 0. Does an inverse exist? 1 yes, 0 no.

    Answer