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Cogito

Linear Algebra · Chapter 6 · Lesson 3

Practice Worksheet

Rank and the Rank-Nullity Theorem

What survives plus what is crushed is always the whole input.

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

NameDate

Figure — use these to answer the problems

Make the second row a multiple of the first. The rank drops from 2 to 1 and the nullity rises from 0 to 1.
1325
  • Determinant of A-1
row 1, column 11
row 1, column 23
row 2, column 12
row 2, column 25
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. 8 columns and rank 5. What is the nullity?

    Answer

  2. What does the rank-nullity theorem say?

    • Rank plus nullity equals the number of columns.
    • Rank always equals nullity.

  3. 6 columns and rank 2. What is the nullity?

    Answer

  4. 4 columns and nullity 1. What is the rank?

    Answer

  5. A 2 by 2 matrix with determinant 0 and non-zero entries. What is its rank?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. The zero matrix with 3 columns. What is its rank?

    Answer

  2. The zero matrix with 3 columns. What is its nullity?

    Answer

  3. A 3 by 3 identity matrix. What is its rank?

    Answer

Stretch Yourself

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

4 problems

  1. Match each quantity to what it measures.

    Draw a line from each item on the left to its match on the right.

    • Rank
    • Nullity
    • Rank plus nullity
    • Independent directions that survive
    • Independent directions crushed to zero
    • The number of columns

  2. 10 columns and rank 10. What is the nullity?

    Answer

  3. The Balance

    A matrix with 7 columns and rank 4. What is its nullity?

    Answer

  4. The Full One

    An invertible 5 by 5 matrix. What is its nullity?

    Answer