study material-mathematics-algebra-matrices

# Shortcut to the Finding the Inverse of a 2X2Matrix

## Shortcut to the Finding the Inverse of a 2×2 Matrix

The inverse of a 2×2 matrix can be found by …

- Switch the elements on the main diagonal
- Take the opposite of the other two elements
- Divide all the values by the determinant of the matrix (since we haven’t talked about the determinant, for a 2×2 system, it is the product of the elements on the main diagonal minus the product of the other two elements).

### Example for the shortcut

Let’s go with an original matrix of

7 | -2 | ||

3 | 5 |

Step 1, switch the elements on the main diagonal would involve switching the 5 and 7.

5 |
-2 | ||

3 | 7 |

Step 2, take the opposite of the other two elements, but leave them where they are.

5 | 2 |
||

-3 |
7 |

Step 3, find the determinant and divide every element by that. The determinant is the product of the elements on the main diagonal minus the product of the elements off the main diagonal. That means the determinant of this matrix is 7(5) - (-3)(2) = 35 + 6 = 41. We divide every element by 41.

The inverse of the original matrix is …

^{5}⁄_{41} |
^{2}⁄_{41} |
||

-^{3}⁄_{41} |
^{7}⁄_{41} |

Now, you’re saying, wait a minute - you said there was no matrix division. There is no division by a matrix. You may multiply or divide a matrix by a scalar (real number) and the determinant is a scalar.