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How to Find the determinant of a 3×3 . matrix

August 18, 2023 by admin Category: How To

You are viewing the article How to Find the determinant of a 3×3 . matrix  at Tnhelearning.edu.vn you can quickly access the necessary information in the table of contents of the article below.

Determinants are fundamental mathematical tools that are used in various areas of mathematics and engineering. In linear algebra, the determinant of a matrix is a scalar value that provides essential information about the properties of the matrix. While finding the determinant of a 2×2 matrix is relatively straightforward, calculating the determinant of a 3×3 matrix involves more complex procedures. In this article, we will explore the process of finding the determinant of a 3×3 matrix step by step, discussing the necessary formulas and techniques along the way. By the end, you will have a solid understanding of how to compute and interpret determinants for 3×3 matrices.

X

wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, 22 people, some of whom are anonymous, have edited and improved the article over time.

This article has been viewed 252,314 times.

The determinant of a matrix is commonly used in calculus, linear algebra, and advanced geometry. Outside the academic world, engineers and computer graphics programmers also have to resort to matrices and their determinants. With this article, the wikiHow will show you how to find the determinant of a 3×3 matrix.

Table of Contents

  • Steps
    • Find the determinant
    • Simplify the math
  • Advice

Steps

Find the determinant

Image titled Find the Determinant of a 3X3 Matrix Step 1

Image titled Find the Determinant of a 3X3 Matrix Step 1

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Write your 3×3 matrix. We will start with a matrix A of size 3×3 and try to find the determinant |A| its. This is the general matrix notation we will be using, and the matrix in the example here is:

  • USA=(a11atwelftha13a21a22a23athirty firsta32a33)=(first53247462){displaystyle M={begin{pmatrix}a_{11}&a_{12}&a_{13}a_{21}&a_{22}&a_{23}a_{31}&a_{32}&a_{33}end{pmatrix }}={begin{pmatrix}1&5&32&4&74&6&2end{pmatrix}}}M={begin{pmatrix}a_{{11}}&a_{{12}}&a_{{13}}a_{{21}}&a_{{22}}&a_{{23}}a_{{31} }&a_{{32}}&a_{{33}}end{pmatrix}}={begin{pmatrix}1&5&32&4&74&6&2end{pmatrix}}
Image titled Find the Determinant of a 3X3 Matrix Step 2

Image titled Find the Determinant of a 3X3 Matrix Step 2

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Select a row or a column. That will be your reference row or column. Either way, the final answer is the same. For now, just pick the first row. Then we’ll have some advice on how to choose the easiest option to calculate.

  • Select the first row of matrix A in our example. Circle row 1 5 3. Or in general terms, circle a 11 a 12 a 13 .
Image titled Find the Determinant of a 3X3 Matrix Step 3

Image titled Find the Determinant of a 3X3 Matrix Step 3

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Cross out rows and columns past your first element. Look at the row or column you’ve circled and select the first element. Draw a line over its row and column. At this point, you have only four numbers left. We will treat them as a 2×2 matrix.

  • In this example, our reference row is 1 5 3. The first element is in row 1 and column 1. Cross out all rows 1 and column 1. Write the remaining elements as a 2×2 matrix. :
  • 1 5 3
    2 4 7
    4 6 2
Image titled Find the Determinant of a 3X3 Matrix Step 4

Image titled Find the Determinant of a 3X3 Matrix Step 4

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Find the determinant of a 2×2 matrix. Remember, the determinant of the matrix (abcd){displaystyle {begin{pmatrix}a&bc&dend{pmatrix}}}{begin{pmatrix}a&bc&dend{pmatrix}} is ad – bc . [1] X Research Source You can remember this formula by drawing an X going from side to side of the 2×2 matrix. Multiply the two numbers connected by the diagonal of X. Next, subtract the product of the two numbers joined by the diagonal /. Use this formula to calculate the determinant of the matrix you just found.

  • In the above example, the determinant of the matrix (4762){displaystyle {begin{pmatrix}4&76&2end{pmatrix}}}{begin{pmatrix}4&76&2end{pmatrix}} = 4 * 2 – 7 * 6 = -34 .
  • This determinant is called a child determinant of the element selected from the original matrix. [2] X Research Source In this case, we have just found the subdeterminance of a 11 .
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Image titled Find the Determinant of a 3X3 Matrix Step 5

Image titled Find the Determinant of a 3X3 Matrix Step 5

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Multiply the determinant you just found by the element you selected. Remember that when specifying rows and columns to cross out, you selected the element from the row (or column) reference. Multiply this element by the determinant you just calculated from the 2×2 matrix.

  • In the above example, we have chosen a 11 , the element has the value 1. Multiply it by -34 (the determinant of the 2×2 matrix), we get 1*-34 = -34 .
Image titled Find the Determinant of a 3X3 Matrix Step 6

Image titled Find the Determinant of a 3X3 Matrix Step 6

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Define your result mark. You will then multiply this result by 1 or -1 to get the algebraic subsection of the selected element. Using 1 or -1 depends on the element’s position in the 3×3 matrix. Memorize this simple sign table to determine the sign of each element:

  • + – +
    – + –
    + – +
  • Since we have chosen a 11 , the element marked a +, we will multiply the result by +1 (in other words, do nothing with the result). That would still be -34 .
  • Or, you can specify the sign with the formula (-1) i+j , where i and j are the row and column of the element, respectively. [3] X Research Sources
Image titled Find the Determinant of a 3X3 Matrix Step 7

Image titled Find the Determinant of a 3X3 Matrix Step 7

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Repeat this process for the second element on your reference row or column. Go back to the original 3×3 matrix and the row or column you circled earlier. Repeat the above process for this element:

  • Cross out the row and column of that element. In our case, the selected element is a 12 (which has a value of 5). Cross out row one (1 5 3) and column two (546){displaystyle {begin{pmatrix}546end{pmatrix}}}{begin{pmatrix}546end{pmatrix}} .
  • View the remaining elements as a 2×2 matrix. Here, it’s the matrix (2742){displaystyle {begin{pmatrix}2&74&2end{pmatrix}}}{begin{pmatrix}2&74&2end{pmatrix}}
  • Find the determinant of this 2×2 matrix. Use the formula ad – bc (2*2 – 7*4 = -24).
  • Multiply by the element selected from the 3×3 matrix. -24 * 5 = -120
  • Determines whether it is necessary to multiply it by -1. Use the sign table or the formula (-1) i+j . Here, the selected element is a 12 , which carries the – sign in the sign table. We have to change the sign of the result: (-1)*(-120) = 120 .
Image titled Find the Determinant of a 3X3 Matrix Step 8

Image titled Find the Determinant of a 3X3 Matrix Step 8

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Repeat with the third element. You have one more algebraic complement to find. Calculate i for the third element in your reference row or column. Here is a quick summary of how to calculate the algebraic complement of element a 13 in this example:

  • Cross out row 1 and column 3 to get (2446){displaystyle {begin{pmatrix}2&44&6end{pmatrix}}}{begin{pmatrix}2&44&6end{pmatrix}}
  • Its determinant is 2*6 – 4*4 = -4.
  • Multiply by element a 13 : -4 * 3 = -12.
  • The element a 13 has the + sign in the sign table, so our answer is -12 .
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Image titled Find the Determinant of a 3X3 Matrix Step 9

Image titled Find the Determinant of a 3X3 Matrix Step 9

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Add the three results. This is the final step. You have just calculated three algebraic complements corresponding to three elements in a single row or column. Add them up and you get the determinant of a 3×3 matrix.

  • In this example, our determinant is -34 + 120 + -12 = 74 .

Simplify the math

Image titled Find the Determinant of a 3X3 Matrix Step 10

Image titled Find the Determinant of a 3X3 Matrix Step 10

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Select the reference row or column with the most zero elements. Remember that you can select any row or column to reference. The answer obtained is unchanged. When selecting rows or columns with multiple zero elements, you only need to calculate the algebraic complement of the non-zero elements. Because:

  • Suppose you select row 2, with the elements a 21 , a 22 , and a 23 . To solve the problem, we have three different 2×2 matrices. Call them A 21 , A 22 , and A 23 respectively.
  • The determinant of a 3×3 matrix is a 21 |A 21 | – a 22 |A 22 | + a 23 |A 23 |.
  • If the elements a 22 and a 23 are both 0, that would be a 21 |A 21 | – 0*|A 22 | + 0*|A 23 | = a 21 |A 21 | – 0 + 0 = a 21 |A 21 |. Now we only have to calculate the algebraic complement of an element.
Image titled Find the Determinant of a 3X3 Matrix Step 11

Image titled Find the Determinant of a 3X3 Matrix Step 11

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Add rows to make the matrix easier. If you take the values of one row and add them to another row, the determinant of the matrix will not change. Similarly, the determinant of the matrix will also remain the same when we do the same with the column. You can manipulate multiple times or multiply the values of the elements of a row or column by a constant before adding them to another row or column to get the maximum number of zeros in the matrix. This saves a lot of computation time.

  • For example, let’s say you have a 3×3 matrix: (9−first23first075−2){displaystyle {begin{pmatrix}9&-1&23&1&07&5&-2end{pmatrix}}}{begin{pmatrix}9&-1&23&1&07&5&-2end{pmatrix}} .
  • To remove the 9 at position a 11 , we can multiply the second row by -3 and add the resulting value to the top row. The new row is [9 -1 2] + [-9 -3 0] = [0 -4 2].
  • The new matrix is (0−423first075−2){displaystyle {begin{pmatrix}0&-4&23&1&07&5&-2end{pmatrix}}}{begin{pmatrix}0&-4&23&1&07&5&-2end{pmatrix}} . Try the same trick with columns to get a 12 to 0.
  • Image titled Find the Determinant of a 3X3 Matrix Step 12

    Image titled Find the Determinant of a 3X3 Matrix Step 12

    {“smallUrl”:”https://www.wikihow.com/images_en/thumb/7/7b/Find-the-Determinant-of-a-3X3-Matrix-Step-12-Version-2.jpg/v4- 728px-Find-the-Determinant-of-a-3X3-Matrix-Step-12-Version-2.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/7/7b/ Find-the-Determinant-of-a-3X3-Matrix-Step-12-Version-2.jpg/v4-728px-Find-the-Determinant-of-a-3X3-Matrix-Step-12-Version-2. jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:”<div class=”mw-parser-output”></div>”}
    Learn how to do quick calculations for triangular matrices. In this special case, the determinant is simply the product of the elements on the main diagonal, from a 11 at the top left to a 33 at the bottom right. Still a 3×3 matrix but with a “triangle” matrix, the other values are not sorted in a special form: [4] X Research Source

    • Upper Triangle Matrix: Every non-zero element belongs to or lies on the main diagonal. Every element below is zero.
    • Lower Triangle Matrix: Every non-zero element belongs to or lies below the main diagonal.
    • Diagonal Matrix: Every non-zero element belongs to the main diagonal (this is a subset of the above two forms).
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  • Advice

    • This method can be extended to all square matrices. For example, if we use it for a 4×4 matrix, the “cross out” step gives us a 3×3 matrix and we can find the determinant of this 3×3 matrix with the steps described above. Be forewarned, though, that hand calculations can get pretty tedious!
    • When every element of a row or a column is 0, the matrix has a determinant of 0.
    X

    wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, 22 people, some of whom are anonymous, have edited and improved the article over time.

    This article has been viewed 252,314 times.

    The determinant of a matrix is commonly used in calculus, linear algebra, and advanced geometry. Outside the academic world, engineers and computer graphics programmers also have to resort to matrices and their determinants. With this article, the wikiHow will show you how to find the determinant of a 3×3 matrix.

    In conclusion, finding the determinant of a 3×3 matrix may seem daunting at first, but with practice and understanding of the steps involved, it can become a manageable task. By using the rule of Sarrus or expanding the matrix using the cofactor method, one can calculate the determinant accurately. The determinant is a useful mathematical concept that has applications in various fields, including physics, engineering, and economics. It provides valuable information about the matrix, such as whether it is invertible or singular. Moreover, the determinant plays a significant role in solving systems of linear equations and finding the area/volume of parallelograms and parallelepipeds. Overall, understanding how to find the determinant of a 3×3 matrix is an essential skill for any mathematician or individual interested in linear algebra.

    Thank you for reading this post How to Find the determinant of a 3×3 . matrix at Tnhelearning.edu.vn You can comment, see more related articles below and hope to help you with interesting information.

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