# Invertible matrix

A matrix

is **invertible** if

so that *A·B = B·A = I _{n}* . Otherwise it is called

**singular matrix**.

*B=A*^{-1} is called the **inverse of A**

* *

NOTE:

Properties:

Let

invertibles, then:

**Calculation of the inverse by Gauss-Jordan method**

To calculate the inverse of an invertible matrix A, we have to transform the matrix (A|I) into the matrix (I|A-1) by using these elementary operations:

- To exchange two rows: R_{i} ↔ R_{j}

- To substitute a row by a linear combination of all the rows: R_{i}↔ k_{1}R_{1}+k_{2}R_{2}+…+k_{i}R_{i}+…k_{m}R_{m} k_{i} ≠ 0, k_{j} are real numbers, j = 1, 2, ….m

** **

Example:

** **

NOTE: if we obtain a row of zeros in the matrix on the left, A is a singular matrix

**Exercise**. Calculate the inverse of the following matrices:

Solutions:

** **

Licensed under the Creative Commons Attribution Non-commercial No Derivatives 3.0 License