Determinant is a fundamental concept in linear algebra used to find a single scalar value for the given matrix. This article will explain what is a 3 × 3 Matrix and how to calculate the Determinant of a 3 × 3 Matrix step by step, as well as, its applications. Whether you are a student learning linear algebra or an enthusiast seeking a deeper understanding of matrix operations, understanding the determinant of a 3 × 3 matrix is a valuable skill to acquire.

What is the Determinant of the Matrix?

Determinant of a Matrix is a single number calculated from a square matrix. In the field of linear algebra, determinants are found by using the values within the square matrix. This number acts like a scaling factor, influencing how the matrix transforms. Determinants are valuable for solving systems of linear equations, finding the inverse of a matrix, and various calculus operations.

What is 3 × 3 Matrix?

A 3 × 3 Matrix is a matrix in which the number of rows and columns both are equal to 3. Since, the number of rows and columns are equal hence 3 × 3 is a square matrix of order 3×3. A matrix is like a table made of numbers, organized into rows and columns. It’s used to store and work with data in mathematics and other fields. Whereas, a 3 × 3 matrix is a specific type of matrix which consists of three rows and three columns. It can be represented as:

3 × 3 Matrix

Properties of 3 × 3 Matrix

Like other matrices, 3 × 3 matrices also have some important properties.

• Square Matrix: A 3 × 3 matrix has three rows and three columns, making it a square matrix.

• Determinant: A 3 × 3 matrix has a determinant, a numerical value crucial for solving equations and finding inverses.

• Matrix Multiplication: You can multiply a 3 × 3 matrix by another matrix if the number of columns in the first matrix matches the number of rows in the second.

• Inverse: A 3 × 3 matrix may have an inverse if its determinant is non-zero. The inverse matrix, when multiplied by the original matrix, yields the identity matrix.

Determinant of 3 × 3 Matrix Formula

There are various methods exist for calculating a matrix’s determinant. The most common approach is by breaking a given 3 × 3 matrix into smaller 2 × 2 determinants. This simplifies the process of finding the determinant and is widely used in linear algebra.

Let’s take a 3 × 3 square matrix which is written as,

To calculate the determinant of matrix A, i.e., |A|.

Expand the Matrix along the elements of first row.

Therefore,

How do you find the Determinant of a 3 × 3 Matrix?

Let us understand the calculation of a 3 × 3 matrix with an example. For the given 3 × 3 matrix below.

[Tex]\begin{bmatrix} 2 & 1 & 3\\ 4 & 0 & 1\\ 2 & -1 & 2 \end{bmatrix} [/Tex]

Step 1: Choose a Reference Row or Column

Select a row and column to start, suppose in this example we take first element (2) as the reference to calculate the determinant of 3 × 3 matrix.

So, expanding along row R₁

Step 2: Cross Out Row and Column

Remove the chosen row and column in order to simplify it in a 2 × 2 matrix.

2×2 Matrix

Step 3: Find the Determinant of the 2 × 2 Matrix

Find the determinant of the 2 × 2 matrix using the formula

Determinant = (a × d) – (b × c)

Cross Multiply

Here, a = 0, b = 1, c = -1, d = 2

putting these values in the above formula of determinant, we get

Determinant = (0 × 2) – (1 × -1)

Determinant = 0- (-1)

Determinant = 0+1

∴ Determinant of the 2 × 2 matrix = 1

Step 4: Multiply by the Chosen Element

Multiply the determinant of the 2 × 2 matrix by the chosen element from the reference row (which is 2,1 and 3 in this case):

first element = 2 × 1 = 2

Step 5: Repeat this process for the second element in the chosen reference row

For Second Element

Find the Determinant for the second element 1 by putting the values of 2×2 matrix in formula

Determinant = (a × d) – (b × c)

Here, a = 4, b= 1, c= 2, d= 2

Determinant = (4 × 2) – (1 × 2)

Determinant = 8 – 2

Determinant = 6

Now, multiply the determinant of the 2 × 2 matrix by the chosen element from the reference row (which is 1 in this case):

second element = 1 × 6 = 6

Step 6: Repeat this process for the third element in the chosen reference row

For Third Element

Find the Determinant for the third element 3 by putting the values of 2×2 matrix in formula

Determinant = (a × d) – (b × c)

Here, a = 4, b= 0, c= 2, d= -1

Determinant = (4 × -1) – (0 × 2)

Determinant = -4 – 0

Determinant = -4

Now, multiply the determinant of the 2×2 matrix by the chosen element from the reference row (which is 3 in this case):

second element = 3 × (-4) = -12

Step 7: Using Formula

Add up all the results from the step 4, 5, and 6

2 – 6 + (-12) = (-16)

∴ -16 is the determinant of the 3 × 3 matrix.

Application of Determinant of a 3 × 3 Matrix

Determinant of a Matrix can be used to find the inverse and solve the system of linear equation. Hence, we learn to find the inverse of 3 × 3 Matrix and also solve system of linear equation using Cramer’s Rule which involve the use of determinant of 3 × 3 Matrix.

Inverse of 3 × 3 Matrix

The formula to find the inverse of a square matrix A is:

[Tex]A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) [/Tex]

Where,

• A-1 is the inverse of matrix A.
• Det(A) represents the determinant of matrix A.
• adj(A) stands for the adjugate of matrix A

In simple terms, you can follow these steps to find the inverse of a matrix:

Step 1. Calculate the determinant of matrix A.

Step 2. Find the adjugate of matrix A.

Step 3. Multiply each element in the adjugate by 1/det(A).

This formula is used for square matrices (matrices with the same number of rows and columns) and assumes that the determinant is non-zero, which is a necessary condition for a matrix to have an inverse.

Cramer’s Rule

Cramer’s Rule provides a formula to solve a system of linear equations using determinants. For a system of linear equations with n variables are given in the form of

AX=B

Where,

• A = Coefficient of the square matrix
• X = Column matrix having variables
• B = Column matrix having constants

Consider the following system of linear equation

a₁x + b₁y + c₁z + . . . = d₁

a₂x + b₂y + c₂z + . . . = d₂

. . .

a_nx + b_ny + c_nz + . . . = d_n

The variables x, y, z, …, are determined using the following formulas:

• x = D_x/D
• y = D_y/D
• z = D_z/D

Where:

• D is the determinant of the coefficient matrix.
• D_x is the determinant of the matrix obtained by replacing the coefficients of x with the constants on the right-hand side.
• D_y is the determinant of the matrix obtained by replacing the coefficients of y
• D_z is the determinant of the matrix obtained by replacing the coefficients of z

Cramer’s Rule is applicable when the determinant of the coefficient matrix D is non-zero. If D = 0, the rule cannot be applied which indicates either no solution or infinitely many solutions depending on the specific case.

Also, Check

• Types of Matrices
• System of Linear Equations with Three Variables
• Matrix Operations

Determinant of 3 × 3 Matrix Solved Examples

Example 1: Find the determinant of matrix A [Tex]\begin{vmatrix} 2 & 3 & 1 \\ 0 & 4 & 5 \\ 1 & 6 & 2 \\ \end{vmatrix} [/Tex]

Determinant of A = 2 (4×2 – 5×6) – 3(0×2 – 5×1) + 1(0×6 – 4×1)

⇒ Determinant of A = 2(8-30) – 3(0-5) +1(0-4)

⇒ Determinant of A =2(-22) – 3(-5) +1(-4)

⇒ Determinant of A = (-44) +15 – 4

⇒ Determinant of A =-44+11

∴ Determinant of A i.e., |A| = (-33)

Example 2: Find determinant of matrix B =[Tex]\begin{vmatrix} 1 & 2 & 1 \\ 0 & 3 & 0 \\ 4 & 1 & 2 \\ \end{vmatrix} [/Tex]

Detrminant of B = 1(3×2 – 0×1) – 2(0×2 – 0×4) + 1(0×1 – 3×4)

⇒ Determinant of B = 1(6-0) – 2(0) + 1(-12)

⇒ Determinant of B = 1(6) – 0 – 12

⇒ Determinant of B =6-12

⇒ Determinant of B = (-6)

∴ Determinant of B i.e., |B| = 6

Example 3: Find determinant of matrix C [Tex]\begin{vmatrix} 3 & 1 & 2 \\ 0 & 2 & 5 \\ 2 & 0 & 4 \\ \end{vmatrix} [/Tex]

Determiinant of matrix C = 3(2×4 – 5×0) – 1(0×4 – 5×2) + 2(0×0 – 2×2)

⇒ Determinant of C = 3(8-0) – 1(0-10) + 2(0-4)

⇒ Determinant of C =3(8) – 1(-10) + 2(-4)

⇒ Determinant of C = 24 + 10 -8

⇒ Determinant of C = 26

∴ Determinant of C i.e., |C| = 26

Example 4: Solve the given system of Equations using Cramer’s Rule

2x + 3y – z = 7
4x – 2y + 3z = 8
x + y + 2z = 10

Solution:

Step1: First, find the Determinant D of coefficient matrix.

[Tex]D = \begin{vmatrix} 2 & 3 & -1 \\ 4 & -2 & 3 \\ 1 & 1 & 2 \end{vmatrix} [/Tex]

On Solving this determinant D

D= 2(-2×2-3×1) – 3(4×2-1×3) – (-1)(4×1-(-2)×3)

⇒ D= 2(-4-3) – 3(8-3) – (-1)(4+6)

⇒ D= 2(-7) – 3(5) – (-1)(10)

⇒ D= -14-15+10

⇒ D= -19

Step2: Now, find the determinants of D_x, D_y and D_z

For D_x, we replace the coefficients of x with the constants on the right-hand side:

[Tex]Dx = \begin{vmatrix} 7 & 3 & -1 \\ 8 & -2 & 3 \\ 10 & 1 & 2 \end{vmatrix} [/Tex]

For D_y, we replace the coefficients of y with the constants:

[Tex]Dy = \begin{vmatrix} 2 & 7 & -1 \\ 4 & 8 & 3 \\ 1 & 10 & 2 \end{vmatrix} [/Tex]

For D_z, we replace the coefficients of z with the constants:

[Tex]Dz = \begin{vmatrix} 2 & 3 & 7 \\ 4 & -2 & 8 \\ 1 & 1 & 10 \end{vmatrix} [/Tex]

On Solving the determinant D_x

D_x = 7(-2×2 – 3×1) – 3(8×2 – 3×10) – (-1)(8×1 – (-2×10)

⇒ D_x = 7(-4 – 3) – 3(16 – 30) – (-1)(8 + 20)

⇒ D_x = 7(-7) – 3(-14) + 28

⇒ D_x = -49 + 42 + 28

Thus, D_x = 21

On Solving the determinant D_y

D_y = 2(-2×2 – 3×10) – 7(4×2 – 1×10) – (-1)(4×1 – (-2×10)

⇒ D_y = 2(-4 – 30) – 7(8 – 10) – (-1)(4 + 20)

⇒ D_y = 2(-34) – 7(-2) + 24

⇒ D_y = -68 + 14 + 24

⇒ D_y = -30

On Solving the determinant D_z

D_z = 2(-2×(-2) – 3×(-2)) – 3(4×(-2) – 1×(-10)) – 7(4×3 – (-2×1)

⇒ D_z = 2(4 + 6) – 3(-8 + 10) – 7(12 + 2)

⇒ D_z = 2(10) – 3(2) – 7(14)

⇒ D_z = 20 – 6 – 98

⇒ D_z = -84

Step 3: Now putting the values of D, D_x, D_y and D_z in the Carmer’s Rule Formula to find the values of x,y and z.

x = D_x/D = 21/(-19)

y = D_y/D = (-30)/(-19)

z = D_z/D = (-84)/(-19)

Practice Questions on Determinant of 3 × 3 Matrix

Q1. Calculate the determinant of the identity matrix:

[Tex]\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} [/Tex]

Q2. Find the determinant of the matrix:

[Tex]\begin{bmatrix} 3 & 2 & 0 \\ 0 & 4 & -1 \\ 2 & 1 & 5 \end{bmatrix} [/Tex]

Q3. Determine the determinant of the matrix:

[Tex]\begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix} [/Tex]

Q4. Calculate the determinant of the matrix:

[Tex]\begin{bmatrix} -1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -3 \end{bmatrix} [/Tex]

Q5. Find the determinant of the matrix:

[Tex]\begin{bmatrix} 4 & 3 & 2 \\ 1 & 0 & 1 \\ 2 & 1 & 4 \end{bmatrix} [/Tex]

Q6. Determine the determinant of the matrix:

[Tex]\begin{bmatrix} 0 & 1 & 2 \\ 2 & -1 & 3 \\ 1 & 0 & -2 \end{bmatrix} [/Tex]

Determinant of 3 × 3 Matrix – FAQs

1. What is A Matrix?

A matrix is a rectangular arrangement of numbers or elements organized in rows and columns. It’s used in various fields to represent and solve mathematical, scientific, and engineering problems.

2. What is the significance of the Determinant of a 3 × 3 Matrix?

The determinant of a 3 × 3 matrix is significant because it provides information about the matrix’s properties. It helps determine if a system of linear equations has a unique solution, among other applications.

3. What is the definition of Determinant of Matrix?

The determinant of a matrix is a scalar value computed from the matrix’s elements, providing information about its properties. It is used in solving systems of linear equations, finding inverses, and more.

4. What if the Determinant of a 3 × 3 Matrix is Zero?

If the determinant of a 3 × 3 matrix is zero, it means that the matrix is singular, and it does not have an inverse. In geometric terms, it indicates that the transformation represented by the matrix collapses the area or volume to zero. determinant is always zero. This is applicable for matrices of any size.

5. Can the Determinant of a 3 × 3 Matrix be Negative?

Yes, the determinant can be negative. The sign of the determinant depends on the arrangement of the matrix elements and whether they result in a positive or negative value according to the calculation method.

6. What are some Practical Applications of finding the Determinant of a 3 × 3 Matrix?

Determinants are used in various fields, including physics, engineering, computer graphics, and economics. They help solve systems of linear equations, analyze geometric transformations, and determine the stability of dynamic systems.