# Investigating Properties of Determinants using Numerical Examples

#### Summary

We use Matlab as an enrichment tool for Elementary Linear Algebra. In this activity, students investigate, numerically, properties of determinants by creating and using random matrices. In particular, student investigate if the determinant of the sum of matrices is the sum of determinants. Is the determinant of the product, the product of determinants? What is the relation between the determinant of a matrix and its inverse? What is the relation between the determinant of a matrix and its transpose? Students also investigate effects of row operations on determinants.

## Learning Goals

The goal of this activity is to investigate properties of determinants. Students use numerical examples to figure out if a property is true for all matrices. For example, Is it true that determinant of the sum of matrices is the sum of determinants? Also, students use numerical examples to find or discover different properties of determinants. For example, what is the relation between the determinant of a matrix and its inverse?

Matlab will help students establish different properties of determinants. Once students are convinced that a given property holds they will try to prove it or justify it using the knowledge that they learned in this course.

Matlab will help students establish different properties of determinants. Once students are convinced that a given property holds they will try to prove it or justify it using the knowledge that they learned in this course.

## Context for Use

Investigating Properties of Determinants using Numerical Examples is a classroom activity designed for Elementary Linear Algebra class. Since students investigate several properties of determinants the classroom activity might take about one hour.

To complete this activity, students need to be familiar with basic Matlab operations, like creating matrices, using special matrices, for example, generating random matrix or creating a random matrix with integer entries. Students will also use matrix operations, like: addition, subtraction, multiplication, transpose, powers, inverse and finding a determinant.

To complete this activity, students need to be familiar with basic Matlab operations, like creating matrices, using special matrices, for example, generating random matrix or creating a random matrix with integer entries. Students will also use matrix operations, like: addition, subtraction, multiplication, transpose, powers, inverse and finding a determinant.

## Description and Teaching Materials

Determine if the following properties are true.

If the statement seems to be true, create at least 3 examples of matrices with integer entries and at least two examples of matrices with random (no integers) entries. Investigate for matrices of size 2, 3, 4, and check for higher orders, like size 10 or 12.

If the statement appears to be not true, create at least two examples.

1. Check if det (A+B) = det A + det B

2. Check if det(AB) = det A det B

3. Compare determinant of a matrix with the determinant of its inverse. What did you notice?

4. Compare determinant of a matrix with the determinant of its transpose.

5. Check if det(3A) = 3detA ? Use matrices with different sizes and use different scalars. Clearly state your conclusion.

6. Investigate effects of row operations on determinants

(a) If two rows of a matrix are interchanged, is the determinant still the same?

(b) If one row is multiplied by a scalar, is the determinant still the same? Use matrices with different sizes. How does that property compare with property that we checked in 5.

(c) If one row is changed by a scalar multiple of another row, is the determinant the same?

Student Handout (Acrobat (PDF) 218kB Aug10 18)

If the statement seems to be true, create at least 3 examples of matrices with integer entries and at least two examples of matrices with random (no integers) entries. Investigate for matrices of size 2, 3, 4, and check for higher orders, like size 10 or 12.

If the statement appears to be not true, create at least two examples.

1. Check if det (A+B) = det A + det B

2. Check if det(AB) = det A det B

3. Compare determinant of a matrix with the determinant of its inverse. What did you notice?

4. Compare determinant of a matrix with the determinant of its transpose.

5. Check if det(3A) = 3detA ? Use matrices with different sizes and use different scalars. Clearly state your conclusion.

6. Investigate effects of row operations on determinants

(a) If two rows of a matrix are interchanged, is the determinant still the same?

(b) If one row is multiplied by a scalar, is the determinant still the same? Use matrices with different sizes. How does that property compare with property that we checked in 5.

(c) If one row is changed by a scalar multiple of another row, is the determinant the same?

Student Handout (Acrobat (PDF) 218kB Aug10 18)

## Teaching Notes and Tips

This activity can be used as an introductory activity when teaching about different properties of determinants. The instructor should first explain what determinant is and show how determinant can be calculated manually using a definition of determinants. Once students understand and are able to calculate determinants, then they can investigate different properties of determinants.

## Assessment

At the end of the class, students discuss and prove all the properties that hold for determinants. If a student can correctly identify each property, then the instructor will know that the goals have been achieved.

We quite often collect students work and grade them as: complete or incomplete assignments.

We quite often collect students work and grade them as: complete or incomplete assignments.