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

• Steps:

1) Put the original matrix and in the left side the corresponding identity matrix (In) in the right side.

2) You compute rref in the left side, keep in mind that the operations also have an effect on the right side.

3) After you have a reduced row echelon form in the left side, the matrix that is left on the right side is the Inverse of the original matrix. 

$ \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

A is a n x n matrix Def: the (i,j) minor of A is the submatrix of A obtained by erasing row i and column j of A. there are n² submissions.