--- title: 3 by 3 Determinants --- ## 3 by 3 Determinants 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.

## Calculation ### Method 1 Consider the following matrix, which we will call A:

a	b	c
d	e	f
g	h	i

Then the determinant of this matrix, denoted det(A), is given by:

det(A) = a * (e * i - h * f) - b * (d * i - f * g) + c * (d * h - e * g)

Please keep in mind the order of operations in the expression above. For example, consider the following matrix, which we will call B:

1	2	3
0	-3	5
-10	4	7

det(B) is given by the formula above. We apply the formula below:

det(B) = 1 * ( (-3) * 7 - 5 * 4) - 2 * ( 0 * 7 - 5 * (-10)) + 3 * (0 * 4 - (-3) * (-10))

, which we simplify to:

det(B) = 1 * ((-21) - 20) - 2 * (0 - (-50)) + 3 * (0 - (30))

, which we simplify to:

det(B) = (-41) - 100 - 90 = -231

### Method 2 - Sarrus's Rule This method is similar to 2 by 2 determinants, and based on opertations with diagonals Again, consider the following matrix, which we will call A:

a	b	c
d	e	f
g	h	i

Then the determinant of this matrix, denoted det(A), is given by:

det(A) = a * e * i + b * f * g + c * d * h - c * e * g - f * h * a - i * b * d

Note how three top-right to bottom-left diagonals are positive

a
	e
		i

	b
		f
g

		c
d
	h

Top-left to bottom-right are negative

		c
	e
g

a
		f
	h

	b
d
		i

Consider the same example as in method 2: matrix, which we will call B:

1	2	3
0	-3	5
-10	4	7

det(B) is given by the formula above. We apply the formula below:

det(B) = 1 * (-3) * 7 + 2 * 5 * (-10) + 3 * 0 * 4 - 3 * (-3) * (-10) - 5 * 4 * 1 - 7 * 2 * 0

, which we simplify to:

det(B) = -21 - 100 + 0 - 90 - 20 - 0 = -231

Which is the same as in method 1 #### 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