Determinant of 3×3 Matrix

A 3 × 3 matrix is a square matrix with three rows and three columns, often used to organize numbers in math and related fields. It plays a key role in linear algebra, mainly when calculating the determinant—a single value that provides essential information about the matrix. This article will explain what a 3 × 3 matrix is, how to find its determinant step by step and explore its practical applications. Whether you're a student or just curious about matrix operations, understanding the determinant is a valuable skill.

An example of a 3 × 3 matrix is shown below:

What is the Determinant of a Matrix?

The determinant of a Matrix is a single number calculated from a square matrix. In 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.

Properties of 3 × 3 Matrix

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

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

There are various methods that 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 the 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.

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

Step 1: Choose a Reference Row or Column

Select a row and column to start, suppose in this example we take the 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 to simplify it in a 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)

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

Find the Determinant for the second element 1 by putting the values of the 2x2 matrix in the 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

Find the Determinant for the third element 3 by putting the values of the 2x2 matrix in the 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 2x2 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 steps 4, 5, and 6

2 - 6 + (-12) = (-16)

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

Also Check:

• Determinant of 2 × 2 Matix
• Determinant of 4 × 4 Matrix

Application of Determinant of a 3 × 3 Matrix

Determinants of a Matrix can be used to find the inverse and solve the system of linear equations. Hence, we learn to find the inverse of a 3 × 3 Matrix and also solve a system of linear equations using Cramer's Rule, which involves the use of the determinant of a 3 × 3 Matrix.

The inverse of 3 × 3 Matrix

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

A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)

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 \begin{vmatrix}2 & 3 & 1 \\0 & 4 & 5 \\1 & 6 & 2 \\\end{vmatrix}

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 the determinant of matrix B =\begin{vmatrix}1 & 2 & 1 \\0 & 3 & 0 \\4 & 1 & 2 \\\end{vmatrix}

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 the Determinant of matrix C \begin{vmatrix}3 & 1 & 2 \\0 & 2 & 5 \\2 & 0 & 4 \\\end{vmatrix}

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.

D = \begin{vmatrix}2 & 3 & -1 \\4 & -2 & 3 \\1 & 1 & 2\end{vmatrix}

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:

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

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

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

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

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

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 Determinants of 3 × 3 Matrix

Question 1: Calculate the determinant of the identity matrix:

\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}

Question 2: Find the determinant of the matrix:

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

Question 3: Determine the determinant of the matrix:

\begin{bmatrix}2 & 1 & 1 \\1 & 2 & 1 \\1 & 1 & 2\end{bmatrix}

Question 4: Calculate the determinant of the matrix:

\begin{bmatrix}-1 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & -3\end{bmatrix}

Question 5: Find the determinant of the matrix:

\begin{bmatrix}4 & 3 & 2 \\1 & 0 & 1 \\2 & 1 & 4\end{bmatrix}

Question 6: Determine the determinant of the matrix:

\begin{bmatrix}0 & 1 & 2 \\2 & -1 & 3 \\1 & 0 & -2\end{bmatrix}

Answer key:

• 1
• 59
• 4
• 6
• -8
• 9