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}2&3|1&0\\4&5|0&1\end{array}\right) $ ----->$ \left(\begin{array}{cccc} 2 & 3 | 1 & 0 \\ 0 & -1 | -2 & 1 \end{array}\right) $------>$ \left(\begin{array}{cccc} 2 & 0 | -5 & 3 \\ 0 & -1 | -2 & 1 \end{array}\right) $ ----> $ \left(\begin{array}{cccc} 1 & 0 | -5/2 & 3/2 \\ 0 & 1 | 2 & -1 \end{array}\right) $

$ A^{-1}=\left(\begin{array}{cccc} -5/2 & 3/2 \\ 2 & -1 \end{array}\right) $

Note: Calculating a Reuduced Row echelon form for a square matrix which n >5 can get complicated and if you get the Reduced row echelon form by consequence you get the Inverse wrong. In some cases