In matrices, determinants are the special numbers calculated from the square matrix. The determinant of a 3 x 3 matrix is calculated for a matrix having 3 rows and 3 columns. The symbol used to represent the determinant is represented by vertical lines on either side, such as | |. The most popular application is to find area of triangle using determinant, where the three vertices of the triangle are considered as the coordinates in an XY plane.

Let A be the matrix, then the determinant of a matrix A is denoted by |A|. To find any matrix such as determinant of 2×2 matrix, determinant of 3×3 matrix, or n x n matrix, the matrix should be a square matrix. It means that the matrix should have an equal number of rows and columns. Finding determinants of a matrix are helpful in solving the inverse of a matrix, a system of linear equations, and so on. In this article, let us discuss how to solve the determinant of a 3×3 matrix with its formula and examples.

Determinant of a 3 x 3 Matrix Formula

We can find the determinant of a matrix in various ways. First, we have to break the given matrix into 2 x 2 determinants so that it will be easy to find the determinant for a 3 by 3 matrix.

Let’s suppose you are given a square matrix C where

C = \(\begin{bmatrix} a & b &c \\ d& e &f \\ g& h &i \end{bmatrix}\)

Let’s calculate the determinant of matrix C,

Det \(\begin{bmatrix} a & b &c \\ d& e &f \\ g& h &i \end{bmatrix}\)

= a. det \(\begin{bmatrix} e & f\\ h & i \end{bmatrix}\) – b.det \(\begin{bmatrix} d & f\\ g & i \end{bmatrix}\) + c . det \(\begin{bmatrix} d & e\\ g & h \end{bmatrix}\)

Few Important points on 3x 3 Determinant Matrix:

• The scalar multipliers to a corresponding 2 x 2 matrix have top row elements a, b and c serving to it.
• The scalar element gets multiplied by 2 x 2 matrix of remaining elements created at the time when vertical and horizontal line segments were drawn through passing through a.
• This is how we construct the 2 by 2 matrices for scalar multipliers b and c.

The determinant of 3 x 3 matrix formula is given by,

\(\begin{bmatrix} a & b &c \\ d& e &f \\ g& h &i \end{bmatrix}\) = \(\begin{bmatrix} 2 & -3 &9 \\ 2 & 0 & -1\\ 1& 4 & 5 \end{bmatrix}\)

Examples

Let us solve some examples to find the determinant of some of the matrices having three rows and columns.

Example 1: Calculate the determinant of the 3 x 3 matrix.

\(\begin{bmatrix} 2 & -3 &1 \\ 2 & 0 & -1\\ 1& 4 & 5 \end{bmatrix}\)

Solution:

Let’s find the correspondence between the generic elements in the formula and elements of real problem.

\(\begin{bmatrix} a & b &c \\ d& e &f \\ g& h &i \end{bmatrix}\) = \(\begin{bmatrix} 2 & -3 &1 \\ 2 & 0 & -1\\ 1& 4 & 5 \end{bmatrix}\)

Use the 3 x 3 determinant formula:

Applying the formula,

= 2[ 0 – (-4)] + 3 [10 – (-1)] +1 [8-0]

= 2 (0+4) +3 (10 +1) + 1(8)

= 2(4) +3(11) + 8

= 8+33+8

= 49

Therefore, the determinant of \(\begin{bmatrix} 2 & -3 &9 \\ 2 & 0 & -1\\ 1& 4 & 5 \end{bmatrix}\) = 49

Example 2: Calculate the determinant of the 3 x 3 matrix.

\(\begin{bmatrix} 1 & 3 &2 \\ -3 & -1 & -3\\ 2& 3 & 1 \end{bmatrix}\)

Solution:

Let’s find the correspondence between the generic elements in the formula and elements of real problem.

\(\begin{bmatrix} a & b &c \\ d& e &f \\ g& h &i \end{bmatrix}\) = \(\begin{bmatrix} 1 & 3 &2 \\ -3 & -1 & -3\\ 2& 3 & 1 \end{bmatrix}\)

Use the 3 x 3 determinant formula:

= 1[ -1 – (-9)] – 3 [-3 – (-6)] + 2 [-9 – (-2)]

= 1 (-1+9) -3 (-3 +6) + 2(-9 + 2)

= 1(8) -3(3) +2(-7)

= 8 -9-14

= -15

Therefore, the determinant of \(\begin{bmatrix} 1 & 3 &2 \\ -3 & -1 & -3\\ 2& 3 & 1 \end{bmatrix}\) =-15

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