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---
title: 3 by 3 Determinants
---
## 3 by 3 Determinants
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3x3 determinants are a value that can be calculated by the values in a matrix. It is also known as the scaling factor of the linear transformation that the matrix represents.
When a 3x3 matrix and its rows are comprised of three vectors, the determinant of this 3x3 matrix is the volume of the parallelepiped that is made up of these three vectors.
< img src = "https://upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Determinant_parallelepiped.svg/950px-Determinant_parallelepiped.svg.png" width = "400" >
## Calculation
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### Method 1
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Consider the following matrix, which we will call A:
< table >
< tr >
< td style = "background-color: white" > a< / td >
< td style = "background-color: white" > b< / td >
< td style = "background-color: white" > c< / td >
< / tr >
< tr >
< td style = "background-color: white" > d< / td >
< td style = "background-color: white" > e< / td >
< td style = "background-color: white" > f< / td >
< / tr >
< tr >
< td style = "background-color: white" > g< / td >
< td style = "background-color: white" > h< / td >
< td style = "background-color: white" > i< / td >
< / tr >
< / table >
Then the determinant of this matrix, denoted < em > det(A)< / em > , is given by:
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< pre >< em > det(A) = a * (e * i - h * f) - b * (d * i - f * g) + c * (d * h - e * g)</ em ></ pre >
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Please keep in mind the order of operations in the expression above.
For example, consider the following matrix, which we will call B:
< table >
< tr >
< td style = "background-color: white" > 1< / td >
< td style = "background-color: white" > 2< / td >
< td style = "background-color: white" > 3< / td >
< / tr >
< tr >
< td style = "background-color: white" > 0< / td >
< td style = "background-color: white" > -3< / td >
< td style = "background-color: white" > 5< / td >
< / tr >
< tr >
< td style = "background-color: white" > -10< / td >
< td style = "background-color: white" > 4< / td >
< td style = "background-color: white" > 7< / td >
< / tr >
< / table >
< em > det(B)< / em > is given by the formula above. We apply the formula below:
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< pre >< em > det(B) = 1 * ( (-3) * 7 - 5 * 4) - 2 * ( 0 * 7 - 5 * (-10)) + 3 * (0 * 4 - (-3) * (-10))</ em ></ pre >
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, which we simplify to:
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< pre >< em > det(B) = 1 * ((-21) - 20) - 2 * (0 - (-50)) + 3 * (0 - (30))</ em ></ pre >
, which we simplify to:
< pre > < em > det(B) = (-41) - 100 - 90 = -231< / em > < / pre >
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### Method 2 - Sarrus's Rule
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This method it similar to 2 by 2 determinants, and based on opertations with diagonals
Again, consider the following matrix, which we will call A:
< table >
< tr >
< td style = "background-color: white" > a< / td >
< td style = "background-color: white" > b< / td >
< td style = "background-color: white" > c< / td >
< / tr >
< tr >
< td style = "background-color: white" > d< / td >
< td style = "background-color: white" > e< / td >
< td style = "background-color: white" > f< / td >
< / tr >
< tr >
< td style = "background-color: white" > g< / td >
< td style = "background-color: white" > h< / td >
< td style = "background-color: white" > i< / td >
< / tr >
< / table >
Then the determinant of this matrix, denoted < em > det(A)< / em > , is given by:
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< pre >< em > det(A) = a * e * i + b * f * g + c * d * h - c * e * g - f * h * a - i * b * d</ em ></ pre >
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Note how three top-right to bottom-left diagonals are positive
< table >
< tr >
< td style = "background-color: white" > a< / td >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > < / td >
< / tr >
< tr >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > e< / td >
< td style = "background-color: white" > < / td >
< / tr >
< tr >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > i< / td >
< / tr >
< / table >
< table >
< tr >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > b< / td >
< td style = "background-color: white" > < / td >
< / tr >
< tr >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > f< / td >
< / tr >
< tr >
< td style = "background-color: white" > g< / td >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > < / td >
< / tr >
< / table >
< table >
< tr >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > c< / td >
< / tr >
< tr >
< td style = "background-color: white" > d< / td >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > < / td >
< / tr >
< tr >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > h< / td >
< td style = "background-color: white" > < / td >
< / tr >
< / table >
Top-left to bottom-right are negative
< table >
< tr >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > c< / td >
< / tr >
< tr >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > e< / td >
< td style = "background-color: white" > < / td >
< / tr >
< tr >
< td style = "background-color: white" > g< / td >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > < / td >
< / tr >
< / table >
< table >
< tr >
< td style = "background-color: white" > a< / td >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > < / td >
< / tr >
< tr >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > f< / td >
< / tr >
< tr >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > h< / td >
< td style = "background-color: white" > < / td >
< / tr >
< / table >
< table >
< tr >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > b< / td >
< td style = "background-color: white" > < / td >
< / tr >
< tr >
< td style = "background-color: white" > d< / td >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > < / td >
< / tr >
< tr >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > < / td >
< td style = "background-color: white" > i< / td >
< / tr >
< / table >
Consider the same example as in method 2: matrix, which we will call B:
< table >
< tr >
< td style = "background-color: white" > 1< / td >
< td style = "background-color: white" > 2< / td >
< td style = "background-color: white" > 3< / td >
< / tr >
< tr >
< td style = "background-color: white" > 0< / td >
< td style = "background-color: white" > -3< / td >
< td style = "background-color: white" > 5< / td >
< / tr >
< tr >
< td style = "background-color: white" > -10< / td >
< td style = "background-color: white" > 4< / td >
< td style = "background-color: white" > 7< / td >
< / tr >
< / table >
< em > det(B)< / em > is given by the formula above. We apply the formula below:
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< pre >< em > det(B) = 1 * (-3) * 7 + 2 * 5 * (-10) + 3 * 0 * 4 - 3 * (-3) * (-10) - 5 * 4 * 1 - 7 * 2 * 0</ em ></ pre >
, which we simplify to:
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< pre > < em > det(B) = -21 - 100 + 0 - 90 - 20 - 0 = -231< / em > < / pre >
Which is the same as in method 1
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#### More information:
* [Determinant of a Matrix ](https://www.mathsisfun.com/algebra/matrix-determinant.html ) on MathIsFun
* [3x3 Determinant calculator ](http://www.wolframalpha.com/widgets/view.jsp?id=7fcb0a2c0f0f41d9f4454ac2d8ed7ad6 )
* [Determinant ](https://en.wikipedia.org/wiki/Determinant ) on Wikipedia