Inverse Matrices

Definition Per definition an inverse matrix satisfies $A^{-1}A=I$. For $A^{-1}A=A{A}^{-1}=I$ to hold, the matrix $A$ must be a square matrix. A matrix $A$ is invertible when

• $A$ must have nonzero pivots after solving by elimination
• The determinant must be nonzero $\det A\ne 0$
• The equation $Ax=0$ must have only one solution, $x=0$ Property Inheritance
• If $A,B$ are of same size and both invertible, $A{B}^{-1}$ also exists

Recognizing Invertible Matrices

Diagonally dominant matrices

Diagonally dominant matrices are invertible. They exhibit the existence of a diagonal value $a\_ii$ which is larger than the sum of all other values in the same row $i$, $\sum \_j\ne i|a\_ij|$. This condition holds for $A$, for which $A^{-1}$ exists.

$$A=\left[\begin{array}{ccc}3& 1& 1\\ 1& 3& 1\\ 1& 1& 3\end{array}\right]$$