Matrix Inverse Algorithm/Examples/Arbitrary Matrix 2

Examples of use of Matrix Inverse Algorithm

Let $\mathbf A$ be the (square) matrix defined as:

$\mathbf A = \begin {pmatrix} 1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & -1 & 0 \\ \end {pmatrix}$

Then its inverse $\mathbf A^{-1}$ is:

$\mathbf A^{-1} = \begin {pmatrix} 0 & -1 & -1 \\ 0 & 0 & -1 \\ -1 & -1 & -1 \\ \end {pmatrix}$

Proof

We construct $\begin {pmatrix} \mathbf A & \mathbf I \end {pmatrix}$:

$\begin {pmatrix} \mathbf A & \mathbf I \end {pmatrix} = \paren {\begin {array} {ccc|ccc} 1 & 0 & -1 & 1 & 0 & 0 \\ -1 & 1 & 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 & 0 & 1 \\ \end {array} }$

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

The matrix that results from having applied $e_1$ to $e_k$ in order is denoted $\begin {pmatrix} \mathbf A_k & \mathbf B_k \end {pmatrix}$.

$e_1 := r_2 \to r_2 + r_1$

Hence:

$\begin {pmatrix} \mathbf A_1 & \mathbf B_1 \end {pmatrix} = \paren {\begin {array} {ccc|ccc} 1 & 0 & -1 & 1 & 0 & 0 \\ 0 & 1 & -1 & 1 & 1 & 0 \\ 0 & -1 & 0 & 0 & 0 & 1 \\ \end {array} }$

$e_2 := r_3 \to r_3 + r_2$

$\begin {pmatrix} \mathbf A_2 & \mathbf B_2 \end {pmatrix} = \paren {\begin {array} {ccc|ccc} 1 & 0 & -1 & 1 & 0 & 0 \\ 0 & 1 & -1 & 1 & 1 & 0 \\ 0 & 0 & -1 & 1 & 1 & 1 \\ \end {array} }$

$e_3 := r_3 \to -r_3$

$\begin {pmatrix} \mathbf A_3 & \mathbf B_3 \end {pmatrix} = \paren {\begin {array} {ccc|ccc} 1 & 0 & -1 & 1 & 0 & 0 \\ 0 & 1 & -1 & 1 & 1 & 0 \\ 0 & 0 & 1 & -1 & -1 & -1 \\ \end {array} }$

$e_4 := r_1 \to r_1 + r_3$

$e_5 := r_2 \to r_2 + r_3$

$\begin {pmatrix} \mathbf A_5 & \mathbf B_5 \end {pmatrix} = \paren {\begin {array} {ccc|ccc} 1 & 0 & 0 & 0 & -1 & -1 \\ 0 & 1 & 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & -1 & -1 & -1 \\ \end {array} }$

and it is seen that $\begin {pmatrix} \mathbf A_5 & \mathbf B_5 \end {pmatrix}$ is the required reduced echelon form:

$\mathbf A_5 = \mathbf I$

and so by the Matrix Inverse Algorithm:

$\mathbf B_5 = \mathbf A^{-1}$

$\blacksquare$