7 Finding inverses using GE

Note: if you are required to find an inverse using minors in this class, do not use any GE, even for attractive simplifications.

Note: If a matrix is orthonormal, its inverse must be found using a transpose in this class.

Finding the inverse of a matrix is usually a bad idea. But if you do need it, you may use the following trick:

1. Create an augmented matrix in which the $n$ right hand side column vectors are the columns of the unit matrix.
2. Reduce the matrix to row canonical form using Gaussian elimination as described earlier. In other words
   
   $$(A \vert I ) \quad \stackrel{\rm GE}{\longrightarrow} \quad (A_{\rm Row Can} \vert B_{\rm Red})$$
   
   

3. For a nonsingular matrix $A$, $A_{\rm Row Can}$ will be a unit matrix and $B_{\rm Red}$ will be the desired $A^{-1}$. (For a singular matrix, one of the pivots will be found to be zero in the reduction to echelon form, and there is no inverse matrix.)

Note that what you are really doing in the above is solving the system of equations $A A^{-1} = I$ for the columns of $A^{-1}$. The only trick is that if you reduce the matrix all the way to a unit matrix, the solution $A^{-1}$ becomes the same as the right hand side of the reduced system.

Also note that the most efficient way to find the inverse, in terms of operations, is still LU decomposition. Normally this would take $\frac13n^3$ operations to find $L$ and $U$ and then $n^3$ operations to find the $n$ columns of $A^{-1}$. But the right hand sides to the $U\vec y=\vec b$ systems are unit vectors, which are mostly zeros. If you use this to avoid computing coefficients in $y$ that are automatically zero, you can reduce the operations by $\frac13n^3$.