Determinant of a 3 x 3 Matrix

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 the area of a triangle using a determinant, where the three vertices of the triangle are considered as the coordinates in an XY plane.

Let A be the matrix, and then the determinant of a matrix A is denoted by |A|. To find the determinant of matrices, the matrix should be a square matrix, such as a determinant of 2×2 matrix, determinant of 3×3 matrix, or n x n matrix. It means the matrix should have an equal number of rows and columns. Finding determinants of a matrix is 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 A where

\begin{array}{l} A = [\begin{array}{c} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}] \end{array}

Let’s calculate the determinant of matrix A, i.e., |A|.

\begin{array}{l} | A | = | \begin{array}{c} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} | \end{array}

By expanding along the first row elements, we get;

\begin{array}{l} | A | = | \begin{array}{c} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} | = a_{11} (- 1)^{1 + 1} | \begin{array}{c} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} | + a_{12} (- 1)^{1 + 2} | \begin{array}{c} a_{21} & a_{23} \\ a_{31} & a_{33} \end{array} | + a_{13} (- 1)^{1 + 3} | \begin{array}{c} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} | \end{array}

This can be written as:

\begin{array}{l} | A | = | \begin{array}{c} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} | = a_{11} | \begin{array}{c} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} | - a_{12} | \begin{array}{c} a_{21} & a_{23} \\ a_{31} & a_{33} \end{array} | + a_{13} | \begin{array}{c} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} | \end{array}

Therefore,

|A| = a₁₁ (a₂₂ a₃₃ – a₂₃ a₃₂) – a₁₂ (a₂₁ a₃₃ – a₂₃ a₃₁) + a₁₃ (a₂₁ a₃₂ – a₂₂ a₃₁)

Writing the determinant of a 3×3 matrix – Simple points

To find the determinant of a 3×3 matrix, first we need to find the minor matrices of any row or column elements. Suppose, we want to find the determinant of the matrix A by expanding the first column elements, then write the determinants of 2×2 matrices obtained by eliminating the corresponding elements’s row & column.

For example,

\begin{array}{l} | A | = | \begin{array}{c} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} | \end{array}

For element a₁₁, we can find the 2×2 scalar matrix as follows.

Determinant of 3x3 matrix

For the element a₂₁, the scalar matrix is obtained as:

Determinant of 3x3 matrix 2 Similarly, for the element a₃₁, the 2×2 scalar matrix is obtained as follows:

Determinant of 3x3 matrix 3

Thus, the determinant of matrix A is written as:

Determinant of 3x3 matrix formula

Solved Examples

Let’s understand the calculation of determinant of 3×3 matrices with the help of solved examples below:

Example 1: Calculate the determinant of the 3×3 matrix given below:

\begin{array}{l} A = [\begin{array}{c} 1 & - 1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{array}] \end{array}

Solution:

Let’s compare the given matrix with the general matrix form of the 3×3 matrix.

That means,

\begin{array}{l} A = [\begin{array}{c} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}] = [\begin{array}{c} 1 & - 1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{array}] \end{array}

Here,

a₁₁ = 1, a₁₂ = -1, a₁₃ = 0

a₂₁ = 2, a₂₂ = 3, a₂₃ = 4

a₃₁ = 0, a₃₂ = 1, a₃₃ = 2

Thus, by applying the determinant of a 3×3 matrix formula, we have

\begin{array}{l} | A | = 1. | \begin{array}{c} 3 & 4 \\ 1 & 2 \end{array} | - (- 1) . | \begin{array}{c} 2 & 4 \\ 0 & 2 \end{array} | + 0. | \begin{array}{c} 2 & 3 \\ 0 & 1 \end{array} | \end{array}

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

= (6 – 4) + (4 – 0)

= 2 + 4

= 6

Example 2:

Find the determinant of the following 3×3 matrix.

\begin{array}{l} P = [\begin{array}{c} 0 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{array}] \end{array}

Solution:

Let’s compare the given matrix with the general matrix form of the 3×3 matrix.

That means,

\begin{array}{l} [\begin{array}{c} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}] = [\begin{array}{c} 0 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{array}] \end{array}

Here,

a₁₁ = 0, a₁₂ = 1, a₁₃ = 1

a₂₁ = 1, a₂₂ = 1, a₂₃ = 0

a₃₁ = 1, a₃₂ = 0, a₃₃ = 1

Now, by applying the determinant of a 3×3 matrix formula, we get;

\begin{array}{l} | A | = 0. | \begin{array}{c} 1 & 0 \\ 0 & 1 \end{array} | - 1. | \begin{array}{c} 1 & 0 \\ 1 & 1 \end{array} | + 1. | \begin{array}{c} 1 & 1 \\ 1 & 0 \end{array} | \end{array}

= 0 -1 [1(1) – 0(1)] + 1[1(0) – 1(1)]

= -1(1) + (-1)

= -1 – 1

= -2

Found this topic interesting, explore more on other mathematical topics on BYJU’S- The Learning App.

Learn more on Matrices
Determinant Formula	Inverse Matrix Formulas
Determinants and Matrices	Determinants for Class 12

MATHS Related Links
Area Of Sphere	Completing The Square
Laws Of Exponents	Inverse
Rolle's Theorem	Area And Circumference Of A Circle
Area Of Rectangle Formula	Face Value Of A Number
Cube Root Of 512	Square Root Of 6

MATHS Related Links

Area Of Sphere

Completing The Square

Laws Of Exponents

Inverse

Rolle's Theorem

Area And Circumference Of A Circle

Area Of Rectangle Formula

Face Value Of A Number

Cube Root Of 512

Square Root Of 6

Determinant of a 3 x 3 Matrix Formula

Solved Examples

Comments

Leave a Comment Cancel reply

FREE TEXTBOOK SOLUTIONS