Inverse MA265F11Walther - Rhea

Inverse of a Matrix

Definition: Let A be a square matrix of order n x n(square matrix). If there exists a matrix B such that

A B = I n = B A

Then B is called the inverse matrix of A.

Conditions

A n x n is invertible (non-singular) if: 

• Ax=0 has a unique solution
• There is a B matrix such that A B = In
• Ax=b has a unique solution for any b---x=A ^{− 1}

Properties

• (AB) ^{− 1} = B ^{− 1}A ^{− 1}
• (A1 A2.....Ar) ^{− 1}=Ar ^{− 1}A'''r − 1 ^{− 1}...A1 ^{− 1}
• (A ^{− 1}) ^{− 1} = A
• (A ^{− 1})^T = (A^T) ^{− 1}

Calculations



$ \left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right) $