Invertible matrix

A matrix

is invertible if

so that A·B = B·A = In . Otherwise it is called singular matrix.

B=A-1 is called the inverse of A





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: Ri ↔ Rj

- To substitute a row by a linear combination of all the rows: Ri↔ k1R1+k2R2+…+kiRi+…kmRm ki ≠ 0, kj are real numbers, j = 1, 2, ….m




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:











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