# Matrix Inverses

### Augmented Matrix:

### Row operations:

0 | 1 | 0 |

1 | 0 | 0 |

0 | 0 | 1 |

2 | 0 | 0 |

0 | 1 | 0 |

0 | 0 | 1 |

1 | 0 | 0 |

2 | 1 | 0 |

0 | 0 | 1 |

0 | 1 | 0 |

1 | 0 | 0 |

0 | 0 | 1 |

2 | 0 | 0 |

0 | 1 | 0 |

0 | 0 | 1 |

1 | 0 | 0 |

2 | 1 | 0 |

0 | 0 | 1 |

The inverse of an \(n \times n\) matrix \(M\) is another \(n \times n\) matrix, denoted \(M^{-1}\) such that
\[ MM^{-1} = M^{-1}M = I \]
where \(I\) is the \(n \times n\) identity matrix. Suppose a square matrix can be reduced to the identity by a sequence of elementary row operations. The inverse of this matrix is the result of applying the same operations, in the same order, to the identity.

Some important facts regarding matrix inverses include:

Some important facts regarding matrix inverses include:

- Not all matrices are invertible. A square matrix is invertible if and only if its determinant is nonzero. Non square matrices do not have inverses, though they may have a right or left inverse.
- A matrix that is equal to its own inverse is called an involution.
- A matrix with integer entries will have an inverse with integer entries if and only if its determinant is -1 or 1.
- The determinant of a matrix and the determinant of its inverse are inversely proportional: \( \det(A^{-1}) = \frac{1}{\det(A)} \) for an invertible matrix \(A\).
- For an invertible matrix \(A\) and scalar \(c\), we have \( (cA)^{-1} = \frac{1}{c}A^{-1} \)
- For invertible matrices \(A\) and \(B\), we have \( (AB)^{-1} = B^{-1}A^{-1} \)
- The inverse of the transpose of a matrix is equal to the transpose of its inverse: \((A^T)^{-1} = (A^{-1})^T\) for an invertible matrix \(A\).

This applet shows one method of finding the inverse of a randomly generated \(3 \times 3\) invertible matrix. Use the controls below the augmented matrix to perform elementary row operations. The process is complete when the left side is reduced to the identity.

On large screens, the elementary matrix of the operation will be displayed, and the operations performed will be recorded next to the augmented matrix.

On large screens, the elementary matrix of the operation will be displayed, and the operations performed will be recorded next to the augmented matrix.

This applet was created using JavaScript and the Raphael library. If you are unable to see the applet, make sure you have JavaScript enabled in your browser. This applet may not be supported by older browsers.