# Finding a Matrix’s Inverse with Elementary Matrices

Recall that an elementary matrix $E$ performs an a single row operation on a matrix $A$ when multiplied together as a product $EA$. If $A$ is an $n \times n$ matrix, then we can say that $A$ is constructed from applying a finite set of elementary row operations on $I_n$. We first take a finite set of elementary matrices $E_1, E_2, …, E_k$ used to reduce $A$ to $I$:

(1)

If we take this equation and multiply the $E_i$‘s by their inverses $E_i^{-1}$ successively on the left, we get that:

(2)

If we take the earlier formula and multiply the equation on the right by $A^{-1}$, it also follows that:

(3)

We can apply these formulas to help us find $A$ or $A^{-1}$ whenever we need it.

# Using Elementary Matrices to Invert a Matrix

Suppose that we have an invertible matrix $A$. If we append $A$ to $I$ in the manner: $[A \mid I ]$ and reduce $A$ to $I$, we will effectively produce $I$ to $A^{-1}$, that is:

(4)

For example, consider the following matrix:

(5)

First let’s append the identity matrix to $A$ such that we obtain $[A \mid I ]$:

(6)

We will proceed with the following row operations to reduce $A$ to $I$ (omitting details), which will result in turning $I$ to $A^{-1}$:

(7)

From this reduction we obtain that $A^{-1} = \begin{bmatrix} -\frac{3}{16} & \frac{7}{16} & \frac{5}{16} \\ -\frac{1}{8} & \frac{5}{8} & -\frac{1}{8}\\ \frac{5}{16} & -\frac{1}{16} & -\frac{3}{16} \end{bmatrix}$.

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