# Determinant of a 3x3 matrix

$3 \times 3$ determinant
Find the determinant of a 3x3 matrix,
, by using the cofactor expansion.
Proof
The cofactor expansion of $A$ along the first column is
Calculating the 2x2 determinant in each term,
We obtain

Examples
Find the determinants of the following matrices.

By the formula of $3 \times 3$ determinant,

Calculator
Enter a 3x3 matrix and press "Execute" button. Then the determinant is displayed.
 1 2 3 1 2 3
$|A|$ =
Rule of Sarrus
The $3 \times 3$ determinant is a bit more complicated than the $2 \times 2$ determinant, so there is a visual formula to remember.
Let us draw five lines from the top left to the bottom right of a $3 \times 3$ matrix.
The product of all the elements passing through the 3rd line is $A_{11}A_{22}A_{33}$. The product of all the elements through the 2nd and 5th lines is $A_{12}A_{23}A_{31}$. The product of all the elements through the 1st and 4th lines is $A_{13}A_{21}A_{32}$. The sum of the products above is
$$\tag{4.1}$$ Next, let us draw five lines from the upper right to the lower left of the matrix.
The product of all the elements passing through the third line is $A_{13}A_{22}A_{31}$. The product of all the elements through the 2nd and 5th lines is $A_{12}A_{21}A_{33}$. The product of all the elements through the 1st and 4th lines is $A_{11}A_{23}A_{32}$ Multiplying the above products by $-1$ and adding them together gives The sum of the products above multiplied by $-1$ is
$$\tag{4.2}$$ By adding $(4.1)$ and $(4.2)$, we have
, which is equal to the determinant of the $3 \times 3$ matrix.
Thus, the determinant of a $3 \times 3$ matrix is obtained by adding the terms obtained by drawing lines from the upper left to the lower right, and subtracting the terms obtained by drawing lines from the upper right to the lower left. This visual formula is called Rule of Sarrus.