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↔ k₁R₁+k₂R₂+…+k_iR_i+…k_mR_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: