Determinant of a 3×3 Matrix
Online calculator to calculate the determinant of a 3x3 matrix
Determinant Calculator
Instructions
Enter the values of the matrix whose determinant is to be calculated. Empty fields are counted as zero. Click Calculate.
Matrix Determinant - Overview
What is a Determinant?
The determinant is a scalar value that can be computed from the elements of a square matrix. It is a useful tool in solving systems of linear equations and has many applications in mathematics and engineering.
Calculation Formula
For a 3×3 matrix, the determinant is calculated using the Rule of Sarrus or cofactor expansion:
\(\displaystyle\det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\)
Example
Given:
\(\displaystyle A = \begin{bmatrix}2 & 2 & 3\\1 & 3 & 4\\1 & 1 & 1\end{bmatrix}\)
Calculation:
\(\det(A) = 2(3·1 - 4·1) - 2(1·1 - 4·1) + 3(1·1 - 3·1)\)
\(= 2(3 - 4) - 2(1 - 4) + 3(1 - 3)\)
\(= 2(-1) - 2(-3) + 3(-2)\)
Result:
\(\det(A) = -2 + 6 - 6 = -2\)
Properties
- Zero determinant: Matrix is singular (not invertible)
- Non-zero determinant: Matrix is invertible
- Product rule: det(AB) = det(A) · det(B)
- Transpose: det(A) = det(AT)
- Scalar multiplication: det(kA) = kndet(A)
Description of the Determinant of a Matrix
Fundamentals
The determinant is a number (a scalar) that is assigned to a square matrix and can be calculated from its elements. It is a useful tool in solving systems of linear equations, determining if a matrix is invertible, and calculating matrix properties.
Rule of Sarrus (3×3):
\(\displaystyle\det\begin{bmatrix}a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33}\end{bmatrix}\)
\(= a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32}\)
\(- a_{13}a_{22}a_{31} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33}\)
Calculation Methods
- Rule of Sarrus: Diagonal multiplication pattern
- Cofactor expansion: Expansion along row or column
- Row operations: Using elementary row transformations
- Properties: Using determinant properties
Detailed Example
Step-by-Step Calculation
Task:
\(\displaystyle\det\begin{bmatrix}2 & 2 & 3\\1 & 3 & 4\\1 & 1 & 1\end{bmatrix}\)
Step 1: Cofactor expansion (first row)
\(= 2\begin{vmatrix}3 & 4\\1 & 1\end{vmatrix} - 2\begin{vmatrix}1 & 4\\1 & 1\end{vmatrix} + 3\begin{vmatrix}1 & 3\\1 & 1\end{vmatrix}\)
Step 2: Calculate 2×2 determinants
\(= 2(3-4) - 2(1-4) + 3(1-3)\)
\(= 2(-1) - 2(-3) + 3(-2)\)
Step 3: Result
\(= -2 + 6 - 6 = -2\)
Important Properties
- Identity matrix: det(I) = 1
- Inverse: det(A-1) = 1/det(A)
- Row swap: Changes sign of determinant
- Row scaling: det(kA) = k3det(A) for 3×3
- Triangular matrix: Product of diagonal elements
- Zero row/column: det = 0
Practical Applications
Mathematics & Physics:
- Solving systems of linear equations (Cramer's rule)
- Testing matrix invertibility
- Calculating areas and volumes
- Finding eigenvalues
Engineering & Computer Science:
- Computer graphics transformations
- Control systems stability analysis
- Structural engineering stress analysis
- Machine learning algorithms
Important Note
A matrix is invertible if and only if its determinant is not zero (det(A) ≠ 0). If det(A) = 0, the matrix is called singular and does not have an inverse. This property is crucial when solving systems of linear equations.
A detailed description of the determinant can be found here