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 the Reuduced Row echelon form for a square matrix with n >5 can get complicated and if you get the Reduced row echelon form wrong by consequence you get the Inverse wrong. In some cases it is better to use the adjacent matrix as I will show on the next section.

Adjacent Matrix