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Table of contents
1.
Introduction
2.
Basics of Determinants
3.
Evaluation of Determinants
3.1.
Evaluation of 2nd Order Determinant
3.2.
Evaluation of 3rd Order Determinant
3.3.
Evaluation of 4th Order Determinant
4.
Properties of Determinants
5.
FAQs
6.
Key Takeaways
Last Updated: Mar 27, 2024

Determinants

Author Rajat Agrawal
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Introduction

Determinants are mathematical objects that have various applications in engineering mathematics. It is a scalar value calculated from the elements of a square matrix.

Determinants are an arrangement of numbers in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced></math>.

Some of the applications of determinants are; we can use them to solve simultaneous equations and evaluate vector products.

In this blog, we will discuss how to calculate the value of determinants and their various properties.

Basics of Determinants

Mathematicians defined the symbol <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced></math> as a determinant of order 2, and the four numbers arranged in row and column are called elements of determinant. In a determinant, horizontal lines are rows, and vertical lines are called columns. The shape of every determinant is a square. 

If a determinant is of order n, it contains n rows and n columns.

Determinant of order 2 is determined by:

  <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mi>a</mi><mi>d</mi><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mi>b</mi><mi>c</mi></math>

Determinant of order 3 is determined by:

 <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd><mtd><mi>c</mi></mtd></mtr><mtr><mtd><mi>d</mi></mtd><mtd><mi>e</mi></mtd><mtd><mi>f</mi></mtd></mtr><mtr><mtd><mi>g</mi></mtd><mtd><mi>h</mi></mtd><mtd><mi>i</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mi>a</mi><mfenced><mrow><mi>e</mi><mo>&#xB7;</mo><mi>i</mi><mo>-</mo><mi>f</mi><mo>&#xB7;</mo><mi>h</mi></mrow></mfenced><mo>-</mo><mi>b</mi><mfenced><mrow><mi>d</mi><mo>&#xB7;</mo><mi>i</mi><mo>-</mo><mi>g</mi><mo>&#xB7;</mo><mi>f</mi></mrow></mfenced><mo>+</mo><mi>c</mi><mfenced><mrow><mi>d</mi><mo>&#xB7;</mo><mi>h</mi><mo>-</mo><mi>g</mi><mo>&#xB7;</mo><mi>e</mi></mrow></mfenced></math>

Note:  a.) The number of elements in a determinant of order n is <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>n</mi><mn>2</mn></msup></math> .

           b.) Determinant of order 1 is the number itself.

Let’s now learn how the evaluation of determinants is done.

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Evaluation of Determinants

We will learn the evaluation of 2nd, 3rd, and 4th order determinants one by one.

Evaluation of 2nd Order Determinant

Determinant of order 2 is determined by:

  <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mi>a</mi><mi>d</mi><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mi>b</mi><mi>c</mi></math>

Examples:

a.) 

 <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mn>10</mn></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>9</mn></mtd><mtd><mn>6</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>10</mn><mo>&#xB7;</mo><mn>6</mn><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mn>5</mn><mo>&#xB7;</mo><mn>9</mn><mspace linebreak="newline"/><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>60</mn><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mn>45</mn><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>15</mn><mspace linebreak="newline"/></math>

b.) 

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mn>7</mn></mtd><mtd><mn>9</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd><mtd><mn>6</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>7</mn><mo>&#xB7;</mo><mn>6</mn><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mn>9</mn><mo>&#xB7;</mo><mn>4</mn><mspace linebreak="newline"/><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>42</mn><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mn>36</mn><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>6</mn><mspace linebreak="newline"/></math>

Evaluation of 3rd Order Determinant

A third-order or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>&#xD7;</mo><mn>3</mn></math> the determinant can be written as:

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd><mtd><mi>c</mi></mtd></mtr><mtr><mtd><mi>d</mi></mtd><mtd><mi>e</mi></mtd><mtd><mi>f</mi></mtd></mtr><mtr><mtd><mi>g</mi></mtd><mtd><mi>h</mi></mtd><mtd><mi>i</mi></mtd></mtr></mtable></mfenced></math>

It can be evaluated with the help of the 2nd order determinant, as shown below.

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd><mtd><mi>c</mi></mtd></mtr><mtr><mtd><mi>d</mi></mtd><mtd><mi>e</mi></mtd><mtd><mi>f</mi></mtd></mtr><mtr><mtd><mi>g</mi></mtd><mtd><mi>h</mi></mtd><mtd><mi>i</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mi>a</mi><mo>&#xB7;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi>e</mi></mtd><mtd><mi>f</mi></mtd></mtr><mtr><mtd><mi>h</mi></mtd><mtd><mi>i</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mi>b</mi><mo>&#xB7;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi>d</mi></mtd><mtd><mi>f</mi></mtd></mtr><mtr><mtd><mi>g</mi></mtd><mtd><mi>i</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mi>c</mi><mo>&#xB7;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi>d</mi></mtd><mtd><mi>e</mi></mtd></mtr><mtr><mtd><mi>g</mi></mtd><mtd><mi>h</mi></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mi>a</mi><mo>&#xB7;</mo><mfenced><mrow><mi>e</mi><mo>&#xB7;</mo><mi>i</mi><mo>-</mo><mi>h</mi><mo>&#xB7;</mo><mi>f</mi></mrow></mfenced><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mi>b</mi><mo>&#xB7;</mo><mfenced><mrow><mi>d</mi><mo>&#xB7;</mo><mi>i</mi><mo>-</mo><mi>g</mi><mo>&#xB7;</mo><mi>f</mi></mrow></mfenced><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mi>c</mi><mo>&#xB7;</mo><mfenced><mrow><mi>d</mi><mo>&#xB7;</mo><mi>h</mi><mo>-</mo><mi>g</mi><mo>&#xB7;</mo><mi>e</mi></mrow></mfenced><mspace linebreak="newline"/></math>

Note: We will put alternating + and - signs.

Examples:

a.) 

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mn>5</mn></mtd><mtd><mn>6</mn></mtd><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mn>8</mn></mtd><mtd><mn>4</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>9</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>5</mn><mo>&#xB7;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>4</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>9</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mn>6</mn><mo>&#xB7;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>8</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>9</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mn>7</mn><mo>&#xB7;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>8</mn></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>5</mn><mo>&#xB7;</mo><mfenced><mrow><mn>4</mn><mo>&#xB7;</mo><mn>9</mn><mo>-</mo><mn>3</mn><mo>&#xB7;</mo><mn>1</mn></mrow></mfenced><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mn>6</mn><mo>&#xB7;</mo><mfenced><mrow><mn>8</mn><mo>&#xB7;</mo><mn>9</mn><mo>-</mo><mn>3</mn><mo>&#xB7;</mo><mn>2</mn></mrow></mfenced><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mn>7</mn><mo>&#xB7;</mo><mfenced><mrow><mn>8</mn><mo>&#xB7;</mo><mn>1</mn><mo>-</mo><mn>2</mn><mo>&#xB7;</mo><mn>4</mn></mrow></mfenced><mspace linebreak="newline"/><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>5</mn><mo>&#xB7;</mo><mfenced><mn>33</mn></mfenced><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mn>6</mn><mo>&#xB7;</mo><mfenced><mn>66</mn></mfenced><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mn>7</mn><mo>&#xB7;</mo><mfenced><mn>0</mn></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>-</mo><mn>231</mn></math>

b.) 

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mo>-</mo><mn>5</mn></mtd><mtd><mn>7</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>8</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>5</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>-</mo><mn>5</mn><mo>&#xB7;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>8</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>5</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mn>7</mn><mo>&#xB7;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>8</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd><mtd><mn>5</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>-</mo><mn>2</mn><mo>&#xB7;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>-</mo><mn>5</mn><mo>&#xB7;</mo><mfenced><mrow><mo>-</mo><mn>5</mn><mo>-</mo><mn>8</mn></mrow></mfenced><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mn>7</mn><mo>&#xB7;</mo><mfenced><mrow><mn>0</mn><mo>-</mo><mn>32</mn></mrow></mfenced><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mn>2</mn><mo>&#xB7;</mo><mfenced><mrow><mn>0</mn><mo>+</mo><mn>4</mn></mrow></mfenced><mspace linebreak="newline"/><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>65</mn><mo>+</mo><mn>224</mn><mo>-</mo><mn>8</mn><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>281</mn><mspace linebreak="newline"/><mspace linebreak="newline"/></math>

Evaluation of 4th Order Determinant

A 4th-order or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>&#xD7;</mo><mn>4</mn></math> the determinant can be written as:

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd><mtd><mi>c</mi></mtd><mtd><mi>d</mi></mtd></mtr><mtr><mtd><mi>e</mi></mtd><mtd><mi>f</mi></mtd><mtd><mi>g</mi></mtd><mtd><mi>h</mi></mtd></mtr><mtr><mtd><mi>i</mi></mtd><mtd><mi>j</mi></mtd><mtd><mi>k</mi></mtd><mtd><mi>l</mi></mtd></mtr><mtr><mtd><mi>m</mi></mtd><mtd><mi>n</mi></mtd><mtd><mi>o</mi></mtd><mtd><mi>p</mi></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/></math>

It can be evaluated using third-order determinants. Again, be careful about the alternating plus and minus sign.

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd><mtd><mi>c</mi></mtd><mtd><mi>d</mi></mtd></mtr><mtr><mtd><mi>e</mi></mtd><mtd><mi>f</mi></mtd><mtd><mi>g</mi></mtd><mtd><mi>h</mi></mtd></mtr><mtr><mtd><mi>i</mi></mtd><mtd><mi>j</mi></mtd><mtd><mi>k</mi></mtd><mtd><mi>l</mi></mtd></mtr><mtr><mtd><mi>m</mi></mtd><mtd><mi>n</mi></mtd><mtd><mi>o</mi></mtd><mtd><mi>p</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mi>a</mi><mo>&#xB7;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi>f</mi></mtd><mtd><mi>g</mi></mtd><mtd><mi>h</mi></mtd></mtr><mtr><mtd><mi>j</mi></mtd><mtd><mi>k</mi></mtd><mtd><mi>l</mi></mtd></mtr><mtr><mtd><mi>n</mi></mtd><mtd><mi>o</mi></mtd><mtd><mi>p</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mi>b</mi><mo>&#xB7;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi>e</mi></mtd><mtd><mi>g</mi></mtd><mtd><mi>h</mi></mtd></mtr><mtr><mtd><mi>i</mi></mtd><mtd><mi>k</mi></mtd><mtd><mi>l</mi></mtd></mtr><mtr><mtd><mi>m</mi></mtd><mtd><mi>o</mi></mtd><mtd><mi>p</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mi>c</mi><mo>&#xB7;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi>e</mi></mtd><mtd><mi>f</mi></mtd><mtd><mi>h</mi></mtd></mtr><mtr><mtd><mi>i</mi></mtd><mtd><mi>j</mi></mtd><mtd><mi>l</mi></mtd></mtr><mtr><mtd><mi>m</mi></mtd><mtd><mi>n</mi></mtd><mtd><mi>p</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mi>d</mi><mo>&#xB7;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi>e</mi></mtd><mtd><mi>f</mi></mtd><mtd><mi>g</mi></mtd></mtr><mtr><mtd><mi>i</mi></mtd><mtd><mi>j</mi></mtd><mtd><mi>k</mi></mtd></mtr><mtr><mtd><mi>m</mi></mtd><mtd><mi>n</mi></mtd><mtd><mi>o</mi></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/></math>

Using the method we learned previously, we can easily calculate the value of these 3rd order determinants generated in the above equation.

Example:

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mn>2</mn></mtd><mtd><mn>4</mn></mtd><mtd><mn>6</mn></mtd><mtd><mn>8</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>3</mn></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>9</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>2</mn><mo>&#xB7;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>3</mn></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>9</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mn>4</mn><mo>&#xB7;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>3</mn></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>9</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mn>6</mn><mo>&#xB7;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>9</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>-</mo><mn>8</mn><mo>&#xB7;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>2</mn><mo>&#xB7;</mo><mfenced><mrow><mn>0</mn><mo>+</mo><mn>81</mn><mo>+</mo><mn>0</mn></mrow></mfenced><mo>-</mo><mn>4</mn><mo>&#xB7;</mo><mfenced><mrow><mn>0</mn><mo>-</mo><mn>99</mn><mo>+</mo><mn>0</mn></mrow></mfenced><mo>+</mo><mn>6</mn><mo>&#xB7;</mo><mfenced><mrow><mn>0</mn><mo>-</mo><mn>33</mn><mo>-</mo><mn>90</mn></mrow></mfenced><mo>-</mo><mn>8</mn><mo>&#xB7;</mo><mfenced><mrow><mn>0</mn><mo>+</mo><mn>0</mn><mo>-</mo><mn>54</mn></mrow></mfenced><mspace linebreak="newline"/><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>162</mn><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mn>396</mn><mo>&#xA0;</mo><mo>-</mo><mn>738</mn><mo>&#xA0;</mo><mo>+</mo><mn>432</mn><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>252</mn><mspace linebreak="newline"/><mo>&#xA0;</mo></math>

Properties of Determinants

Some of the valuable properties of determinants are:-

1.) The determinant's value remains unchanged if the rows and columns are interchanged.

2.) If any two rows or columns of a determinant are interchanged, the sign of the determinant changes.

3.) The determinant is 0 if any two rows or columns of the determinant are the same.

4.) If a variable k is multiplied by any row or column of the determinant, its value is multiplied by k.

5.) If some or all row or column elements can be expressed as the sum of two or more terms, then determinants can also be expressed as the sum of two or more determinants.

FAQs

1. What is Determinant?

A determinant is a scalar value evaluated from the elements of a square matrix.

They are an arrangement of numbers in the form.

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mi>a</mi><mi>d</mi><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mi>b</mi><mi>c</mi></math>

2. List some applications of Determinants.

Some of the applications of determinants are; we can use them to solve simultaneous equations and evaluate vector products.

3. If any two rows or columns of a determinant are the same, how to calculate the value of the determinant

The value of the determinant is 0 if any two rows or columns of the determinant are the same.

Key Takeaways

In this article, we have extensively discussed the Determinants, evaluation of determinants, and different properties of determinants.

We hope that this blog has helped you enhance your knowledge regarding Linear Algebra. If you want to learn more, check out our articles on the Matrix and Types of Matrix.

Check out this problem - Matrix Median

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