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Determinants play a crucial role in linear algebra as they provide valuable information about a given matrix. They allow us to solve systems of linear equations, determine the invertibility of a matrix, and calculate the area or volume of geometric shapes. In this guide, we will focus specifically on 3×3 matrices and explore the process of finding their determinants. Understanding how to compute the determinant of a 3×3 matrix is a fundamental skill for anyone studying linear algebra or working with matrices in various fields such as physics, engineering, or computer science.
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.
Steps
Find the determinant
 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}}}
 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} .
 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 5 324 746 2
 In the above example, the determinant of the matrix (4762){displaystyle {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} .
 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 .
 + – +
– + –
+ – +  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}
 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}}} .
 View the remaining elements as a 2×2 matrix. Here, it’s the matrix (2742){displaystyle {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 .
 Cross out row 1 and column 3 to get (2446){displaystyle {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 .
 In this example, our determinant is 34 + 120 + 12 = 74 .
Simplify the math
 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.
 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}}} .
 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}}} . Try the same trick with columns to get a _{12} to 0.
 Upper Triangle Matrix: Every nonzero element belongs to or lies on the main diagonal. Every element below is zero.
 Lower Triangle Matrix: Every nonzero element belongs to or lies below the main diagonal.
 Diagonal Matrix: Every nonzero element belongs to the main diagonal (this is a subset of the above two forms).
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.
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 is a straightforward process that involves evaluating the sum of the product of the diagonals of the matrix. By following the stepbystep method of expanding the matrix and calculating the determinants of smaller matrices, one can easily find the determinant of a 3×3 matrix. It is important to remember that the determinant is a valuable tool in linear algebra, as it provides information about the matrix’s invertibility, the number of solutions in a system of linear equations, and the volume of parallelepipeds in threedimensional space.
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