Identity Matrix from Upper Triangular Matrix/Examples/Arbitrary Matrix 1

Example of Use of Identity Matrix from Upper Triangular Matrix

Let $\mathbf A$ be the matrix defined as:

$\mathbf A = \begin {bmatrix} 1 & 1 & 1 & 1 \\ 0 & 2 & 2 & 2 \\ 0 & 0 & 3 & 4 \\ \end {bmatrix}$

is row equivalent to the reduced echelon matrix:

$\mathbf E = \begin {bmatrix} 1 & 0 & 0 & -\dfrac 1 2 \\ 0 & 1 & 0 & \dfrac 1 6 \\ 0 & 0 & 1 & \dfrac 4 3 \\ \end {bmatrix}$

Proof

In the following, $\sequence {e_n}_{n \mathop \ge 1}$ denotes the sequence of elementary row operations that are to be applied to $\mathbf A$.

The matrix that results from having applied $e_1$ to $e_k$ in order is denoted $\mathbf A_k$.

$e_1 := r_2 \to \dfrac {r_2} 2$

$e_2 := r_3 \to \dfrac {r_3} 3$

Hence:

$\mathbf A_3 = \begin {bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & \dfrac 3 2 \\ 0 & 0 & 1 & \dfrac 4 3 \\ \end {bmatrix}$

$e_3 := r_1 \to r_1 - r_3$

$e_4 := r_2 \to r_2 - r_3$

$\mathbf A_4 = \begin {bmatrix} 1 & 1 & 0 & -\dfrac 1 3 \\ 0 & 1 & 0 & \dfrac 1 6 \\ 0 & 0 & 1 & \dfrac 4 3 \\ \end {bmatrix}$

$e_5 := r_1 \to r_1 - r_2$

$\mathbf A_5 = \begin {bmatrix} 1 & 0 & 0 & -\dfrac 1 2 \\ 0 & 1 & 0 & \dfrac 1 6 \\ 0 & 0 & 1 & \dfrac 4 3 \\ \end {bmatrix}$

and it is seen that $\mathbf A_5$ is the required reduced echelon form.

$\blacksquare$