The determinant of a 2 x 2 matrix A = \pmatrix{ a & b \cr c & d } is \det(A) = ad-bc .

The inverse of A is A^{-1} = \displaystyle \frac{1}{ad-bc} \pmatrix{ d & -b \cr -c & a }

If the determinant is zero (ie., the matrix is singular) then the matrix does not have an inverse.

The inverse of A is A^{-1} = \displaystyle \frac{1}{ad-bc} \pmatrix{ d & -b \cr -c & a }

If the determinant is zero (ie., the matrix is singular) then the matrix does not have an inverse.

## Summary/Background

The inverse of a matrix does not exist if the determinant is zero. Matrices whose determinant is not zero are called non-singular. Otherwise they are singular. To find the inverse of a 2x2 matrix:

- interchange the elements in the leading diagonal
- change the sign of the other two elements
- divide by the determinant

## Software/Applets used on this page

## Glossary

### matrix

a rectangular or square grid of numbers.

### non-singular

A matrix whose determinant is non-zero.

### union

The union of two sets A and B is the set containing all the elements of A and B.

## This question appears in the following syllabi:

Syllabus | Module | Section | Topic | Exam Year |
---|---|---|---|---|

AQA A-Level (UK - Pre-2017) | FP4 | Matrix algebra | Inverses | - |

AQA AS Further Maths 2017 | Pure Maths | Matrix Determinants and Inverses | Determinants and Inverses | - |

AQA AS/A2 Further Maths 2017 | Pure Maths | Matrix Determinants and Inverses | Determinants and Inverses | - |

CBSE XII (India) | Algebra | Matrices | Inverse of matrices and proof of uniqueness, if it exists. | - |

CCEA A-Level (NI) | FP1 | Matrix algebra | Inverses | - |

Edexcel A-Level (UK - Pre-2017) | FP1 | Matrix algebra | Inverses | - |

Edexcel AS Further Maths 2017 | Core Pure Maths | Matrices | Determinants and Inverses | - |

Edexcel AS/A2 Further Maths 2017 | Core Pure Maths | Matrices | Determinants and Inverses | - |

Methods (UK) | M5 | Matrix algebra | Inverses | - |

OCR A-Level (UK - Pre-2017) | FP1 | Matrix algebra | Inverses | - |

OCR AS Further Maths 2017 | Pure Core | Determinants, Inverses and Equations | Determinants and Inverses | - |

OCR MEI AS Further Maths 2017 | Core Pure A | Determinants and Inverses | Determinants and Inverses | - |

OCR-MEI A-Level (UK - Pre-2017) | FP1 | Matrix algebra | Inverses | - |

Pre-Calculus (US) | E2 | Matrix algebra | Inverses | - |

Scottish Advanced Highers | M3 | Matrix algebra | Inverses | - |

Scottish (Highers + Advanced) | AM3 | Matrix algebra | Inverses | - |

Universal (all site questions) | M | Matrix algebra | Inverses | - |

WJEC A-Level (Wales) | FP1 | Matrix algebra | Inverses | - |