How to Find the Inverse of a Matrix?

The method for finding the inverse of a matrix depends on the size of the matrix. The simplest method is for a 2x2 matrix.

Inverse of a 2x2 Matrix

The inverse of a 2x2 matrix can be found using a specific formula.

The Formula

For a 2x2 matrix represented as:

A = | a  b |
    | c  d |

The inverse of A, denoted as A^-1, is calculated as follows:

*A^-1 = 1 / (ad - bc) | d -b |
| -c a |**

Where:

• ad - bc is the determinant of the matrix.
• ad - bc ≠ 0, otherwise the inverse does not exist.
• The positions of a and d are swapped.
• The signs of b and c are negated.

Steps to calculate:

1. Calculate the determinant (ad - bc): This value must not be zero for the inverse to exist.
2. Swap a and d: Change the positions of elements in the diagonal.
3. Negate b and c: Change the sign of elements not in the diagonal.
4. Multiply the resulting matrix by 1/(ad-bc): Divide each element of the modified matrix by the determinant.

Example

Let’s find the inverse of the matrix:

A = | 2  1 |
    | 4  3 |

1. Determinant: (2 3) - (1 4) = 6 - 4 = 2
2. Swap a and d:

       | 3  1 |
       | 4  2 |

3. Negate b and c:

      | 3 -1 |
      |-4  2 |

4. Multiply by 1/determinant (1/2):

       | 3/2  -1/2 |
       |-4/2  2/2 |

   Which simplifies to:

      | 1.5  -0.5 |
      | -2   1   |

Therefore, the inverse of matrix A is:

A^-1 = | 1.5  -0.5 |
      | -2   1   |

Key takeaways:

• The core process involves swapping diagonal elements, negating non-diagonal elements, and scaling by the reciprocal of the determinant.
• A zero determinant means the matrix is singular, and it does not have an inverse.