Transformation of Unit Matrix into Inverse

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Theorem

Let $\mathbf A$ be a square matrix of order $n$ of the matrix space $\map {\MM_\R} n$.


Let $\mathbf I$ be the unit matrix of order $n$.

Suppose there exists a sequence of elementary row operations that reduces $\mathbf A$ to $\mathbf I$.

Then $\mathbf A$ is invertible.

Futhermore, the same sequence, when performed on $\mathbf I$, results in the inverse of $\mathbf A$.


Proof

For ease of presentation, let $\breve {\mathbf X}$ be the inverse of $\mathbf X$.

We have that $\mathbf A$ can be transformed into $\mathbf I$ by a sequence of elementary row operations.

By repeated application of Elementary Row Operations as Matrix Multiplications, we can write this assertion as:

\(\ds \mathbf E_t \mathbf E_{t - 1} \cdots \mathbf E_2 \mathbf E_1 \mathbf A\) \(=\) \(\ds \mathbf I\)

From Elementary Row Matrix is Invertible:

$\mathbf E_1, \dotsc, \mathbf E_t \in \GL {n, \R}$


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We can multiply on the left both sides of this equation by:

\(\ds \breve {\mathbf E}_1 \breve {\mathbf E}_2 \cdots \breve {\mathbf E}_{t - 1} \breve {\mathbf E}_t \mathbf E_t \mathbf E_{t - 1} \cdots \mathbf E_2 \mathbf E_1 \mathbf A\) \(=\) \(\ds \breve {\mathbf E}_1 \breve {\mathbf E}_2 \cdots \breve {\mathbf E}_{t - 1} \breve {\mathbf E}_t \mathbf I\)
\(\ds \leadsto \ \ \) \(\ds \mathbf {I I} \cdots \mathbf {I I A}\) \(=\) \(\ds \breve {\mathbf E}_1 \breve {\mathbf E}_2 \cdots \breve {\mathbf E}_{t - 1} \breve {\mathbf E}_t \mathbf I\) Definition of Inverse Matrix
\(\ds \leadsto \ \ \) \(\ds \mathbf A\) \(=\) \(\ds \breve {\mathbf E}_1 \breve {\mathbf E}_2 \cdots \breve {\mathbf E}_{t - 1} \breve {\mathbf E}_t\) Definition of Unit Matrix
\(\ds \leadsto \ \ \) \(\ds \breve {\mathbf A}\) \(=\) \(\ds \breve {\breve {\mathbf E} }_t \breve {\breve {\mathbf E} }_{t - 1} \cdots \breve {\breve {\mathbf E} }_2 \breve {\breve {\mathbf E} }_1\) Inverse of Matrix Product, Leibniz's Law
\(\ds \) \(=\) \(\ds \mathbf E_t \mathbf E_{t - 1} \cdots \mathbf E_2 \mathbf E_1\) Inverse of Group Inverse
\(\ds \) \(=\) \(\ds \mathbf E_t \mathbf E_{t - 1} \cdots \mathbf E_2 \mathbf E_1 \mathbf I\) Definition of Unit Matrix


By repeated application of Elementary Row Operations as Matrix Multiplications, each $\mathbf E_n$ on the right hand side corresponds to an elementary row operation.

Hence the result.

$\blacksquare$


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