Regular matrix . It is said of the square matrix whose determinant is different from 0.
Summary
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- 1 Definition
- 2 Properties
- 3 Characterization
- 4 See also
- 5 Sources
Definition
Let A be a square matrix of dimension n , it is said to be regular if there is another matrix B of the same dimension such that the product A · B and B · A is the identity matrix of dimension n . This matrix B is called the inverse matrix of A and is denoted by A -1 .
Example: the identity matrix is a regular matrix.
Properties
- A matrix is regular if, and only if, its determinant is other than 0.
Characterization
Let A be a matrix of dimension n , the following conditions are equivalent (all conditions occur simultaneously or none occur):
- Ais regular (invertible)
- Any system of linear equations (SEL) with matrix of coefficients A, AX = b , is determined compatible.
- Homogeneous SEL AX = 0is determined compatible.
- The rangeof A is n .
- The reduced echelon form of Ais identity.
- Ais the product of elementary matrices.
