Invert a 3×3 Matrix

Online calculator for inverting a 3x3 matrix

Inverse Matrix Calculator

Instructions

Enter the values of the matrix to be inverted. Empty fields are counted as zero. Click Calculate to get the inverse matrix.

Matrix A
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Result: A-1
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Matrix Inversion - Overview

Prerequisites

A matrix can only be inverted if its determinant is not zero. If det(A) = 0, the matrix is singular and has no inverse.

Inversion Formula

For a matrix A, the inverse A-1 satisfies:

\(A \cdot A^{-1} = A^{-1} \cdot A = I\)

where I is the identity matrix.

Example (2×2 Matrix)

Given:

\(\displaystyle A = \begin{bmatrix}a & b\\c & d\end{bmatrix}\)

Inverse formula (Cramer's rule):

\(\displaystyle A^{-1} = \frac{1}{ad-bc}\begin{bmatrix}d & -b\\-c & a\end{bmatrix}\)

Important Properties
  • Existence: A-1 exists only if det(A) ≠ 0
  • Uniqueness: If A-1 exists, it is unique
  • Product rule: (AB)-1 = B-1A-1
  • Transpose: (AT)-1 = (A-1)T
  • Inverse of inverse: (A-1)-1 = A


Description of Matrix Inversion

Cramer's Rule

There is a quick way to get an inverse for a 2 × 2 matrix. This is a special case of Cramer's rule, which is used to solve systems of equations.

Inverse of 2×2 Matrix:

\(\displaystyle \begin{bmatrix}a & b \\ c & d \end{bmatrix}^{-1} = \frac{1}{ad-bc}\begin{bmatrix}d & -b\\-c & a\end{bmatrix}\)

There are three steps to inverting a 2×2 matrix:

  1. Swap the diagonal elements (a ↔ d)
  2. Change the sign of the other elements (-b, -c)
  3. Divide each element by the determinant (ad-bc)
Calculation Methods
  • Cramer's rule: For 2×2 matrices (efficient)
  • Adjugate method: For 3×3 matrices
  • Gauss-Jordan elimination: For larger matrices
  • Row reduction: Augmented matrix [A|I] → [I|A-1]

When Inversion is Not Possible

Singular Matrices

Assume \(ad = bc\) in the formula above. Then we get \(ad - bc = 0\), and we would try to divide by zero.

So there is consequently no inverse. In this case the original matrix A is called a singular matrix.

Another way to get \(ad = bc\) is if the second row of the matrix is a multiple of the first. This indicates linear dependence between rows.

The Determinant

Without actually calculating the inverse matrix, one can decide whether an inverse exists by simply calculating a single number, the denominator in the formula. This denominator is called the determinant.

Key Rule:
  • If det(A) ≠ 0: Matrix is invertible (non-singular)
  • If det(A) = 0: Matrix is singular (no inverse)

The Cramer's rule also exists for larger matrices, but is computationally very inefficient. It is therefore helpful, especially for large matrices, to be able to determine whether the inverse exists before starting. This can be done by defining the determinant of the matrix for large matrices as well.

Practical Applications

Mathematics & Physics:

  • Solving systems of linear equations
  • Finding matrix transformations
  • Calculating least squares solutions
  • Determining coordinate transformations

Engineering & Computer Science:

  • Computer graphics (inverse transformations)
  • Cryptography and encoding
  • Control systems design
  • Machine learning (neural networks)
Computational Efficiency

For large matrices, Cramer's rule becomes computationally expensive. Modern algorithms like Gauss-Jordan elimination or LU decomposition are much more efficient for matrices larger than 3×3. Always check the determinant first to verify invertibility before attempting inversion.