Identity Matrix from Upper Triangular Matrix

Theorem
Let $\mathbf A = \sqbrk a_{m n}$ be an upper triangular matrix of order $m \times n$ with no zero diagonal elements.

Let $k = \min \set {m, n}$.

Then $\mathbf A$ can be transformed into a matrix such that the first $k$ rows and columns form the unit matrix of order $k$.

Proof
By definition of $k$:
 * if $\mathbf A$ has more rows than columns, $k$ is the number of columns of $\mathbf A$.
 * if $\mathbf A$ has more columns than rows, $k$ is the number of rows of $\mathbf A$.

Thus let $\mathbf A'$ be the square matrix of order $k$ consisting of the first $k$ rows and columns of $\mathbf A$:


 * $\mathbf A' = \begin {bmatrix}

a_{1 1} & a_{1 2} & a_{1 3} & \cdots &    a_{1, k - 1} &      a_{1 k} \\ 0 & a_{2 2} & a_{2 3} & \cdots &    a_{2, k - 1} &      a_{2 k} \\ 0 &      0 & a_{3 3} & \cdots &     a_{3, k - 1} &      a_{3 k} \\ \vdots & \vdots &  \vdots & \ddots &           \vdots &      \vdots  \\ 0 &      0 &       0 & \cdots & a_{k - 1, k - 1} & a_{k - 1, k} \\ 0 &      0 &       0 & \cdots &                0 &      a_{k k} \\ \end {bmatrix}$

$\mathbf A$ can be transformed into echelon form $\mathbf B$ by using the elementary row operations:


 * $\forall j \in \set {1, 2, \ldots, k}: e_j := r_j \to \dfrac 1 {a_{j j} } r_j$

Again, let $\mathbf B'$ be the square matrix of order $k$ consisting of the first $k$ rows and columns of $\mathbf B$:


 * $\mathbf B' = \begin {bmatrix}

1 & b_{1 2} & b_{1 3} & \cdots &    b_{1, k - 1} &      b_{1 k} \\ 0 &      1 & b_{2 3} & \cdots &     b_{2, k - 1} &      b_{2 k} \\ 0 &      0 &       1 & \cdots &     b_{3, k - 1} &      b_{3 k} \\ \vdots & \vdots &  \vdots & \ddots &           \vdots &      \vdots  \\ 0 &      0 &       0 & \cdots &                1 & b_{k - 1, k} \\ 0 &      0 &       0 & \cdots &                0 &            1 \\ \end {bmatrix}$

$\mathbf B$ is then transformed into reduced echelon form $\mathbf C$ by means of the elementary row operations:


 * $\forall j \in \set {1, 2, \ldots, k - 1}: e_{j k} := r_j \to r_j - b_{j k} r_k$


 * $\forall j \in \set {1, 2, \ldots, k - 2}: e_{j, k - 1} := r_j \to r_j - b_{j, k - 1} r_{k - 1}$

and so on, until:


 * $e_{1 2} := r_1 \to r_1 - b_{1 2} r_2$

Again, let $\mathbf C'$ be the square matrix of order $k$ consisting of the first $k$ rows and columns of $\mathbf C$:


 * $\mathbf C' = \begin {bmatrix}

1 &     0 &      0 & \cdots &      0 &      0 \\ 0 &     1 &      0 & \cdots &      0 &      0 \\ 0 &     0 &      1 & \cdots &      0 &      0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 &     0 &      0 & \cdots &      1 &      0 \\ 0 &     0 &      0 & \cdots &      0 &      1 \\ \end {bmatrix}$

By inspection, $\mathbf C$ is seen to be the unit matrix of order $k$.