Determinant of 3×3 Matrix (original) (raw)

Last Updated : 30 Aug, 2025

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:

3x3

3 × 3 Matrix

Table of Content

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.

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

Determinant of 3x3 Matrix Formula - 1

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

aaa

Expand the Matrix along the elements of the first row.

Determinant of 3x3 Matrix Formula - 2

Therefore,

Determinant of 3x3 Matrix Formula - 3

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

remove

**Step 2: Cross Out Row and Column

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

Determinant of a 3x3 Matrix step 2

2x2 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)

Determinant of a 3x3 Matrix step 3

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

Screenshot-2023-10-31-134627

For Second Element

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

Screenshot-2023-10-31-135542

For Third Element

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:

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,

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,

Consider the following system of linear equation

a1x + b1y + c1z + . . . = d1
a2x + b2y + c2z + . . . = d2
...
anx + bny + cnz + . . . = dn

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

Where:

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

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)×1)
⇒ D= 2(-4-3) - 3(8-3) - 1(4+2)
⇒ D= 2(-7) - 3(5) - 1(6)
⇒ D= -14 - 15 - 6
⇒ D= -35

**Step2: Now, find the determinants of Dx, Dy and Dz

For Dx, 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 Dy, we replace the coefficients of y with the constants:

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

For Dz, 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 Dx

Dx = 7(-2×2 - 3×1) - 3(8×2 - 3×10) - (-1)(8×1 - (-2×10)
⇒ Dx = 7(-4 - 3) - 3(16 - 30) - -1(8 + 20)
⇒ Dx = 7(-7) - 3(-14) - 28
⇒ Dx = -49 + 42 - 28

Thus, Dx = - 35

On Solving the determinant Dy

Dy = 2(8 × 2 - 3×10) - 7(4×2 - 3×1) - (-1)(4×10 - 8×1)

⇒ Dy = 2(16 - 30) - 7(8 - 3) - 1(40 - 8)
⇒ Dy = 2(-14) - 7(8 - 3) + 1(32)
⇒ Dy = -28 - 35 -32
⇒ Dy = - 95

On Solving the determinant Dz

Dz = 2(-2×(10) - 8×(1)) - 3(4×(10) - 8×(1)) - 7(4×1 - (-2×1)

⇒ Dz = 2(-20 - 8) - 3(40 - 8) - 7(4 + 2)
⇒ Dz = 2(-28) - 3(32) - 7(6)
⇒ Dz = -56- 96 + 42
⇒ Dz = - 110

**Step 3: Now putting the values of D, Dx, Dy and Dz in the Carmer's Rule Formula to find the values of x,y and z.

x = Dx/D = (-35)/(-35 )= 1
y = Dy/D = (-95)/(-35) = 19 /7
z = Dz/D = (-110)/(-35) = 22/7

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: