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Cogito

Linear Algebra · Chapter 5 · Lesson 2

Practice Worksheet

Properties of Determinants

Rules that let you avoid most of the arithmetic.

12 problems · about 23 minutes · N-VM.C.12

NameDate

Figure — use these to answer the problems

Zero out the bottom-left entry and the determinant becomes just the product of the diagonal. Check that against the readout.
4305
  • Determinant of A20
row 1, column 14
row 1, column 23
row 2, column 10
row 2, column 25
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. A triangular matrix with diagonal 5, 2, 3. What is the determinant?

    Answer

  2. Which row operation leaves the determinant unchanged?

    • Adding a multiple of one row to another.
    • Swapping two rows.

  3. A triangular matrix with diagonal 3, 4. What is the determinant?

    Answer

  4. A determinant of 5. Row 1 is multiplied by 3. What is the new determinant?

    Answer

  5. A determinant of 8. Twice row 1 is added to row 2. What is the new determinant?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. A determinant of 6. What is the determinant of the transpose?

    Answer

  2. Determinants 3 and 0. What is the determinant of the product?

    Answer

  3. A matrix with a whole row of zeros. What is its determinant?

    Answer

Stretch Yourself

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

4 problems

  1. Match each row operation to its effect on the determinant.

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

    • Swap two rows
    • Multiply a row by k
    • Add a multiple of one row to another
    • The determinant changes sign
    • The determinant is multiplied by k
    • The determinant is unchanged

  2. A triangular matrix with diagonal 2, 0, 9. What is the determinant?

    Answer

  3. The Product

    Determinants 4 and 9. What is the determinant of the product matrix?

    Answer

  4. The Swap

    A determinant of 7. Two rows are swapped. What is the new determinant?

    Answer