Matrix Inverse Algorithm/Examples

Examples of use of Matrix Inverse Algorithm

Arbitrary Matrix $1$

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

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

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

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

Arbitrary Matrix $2$

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

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

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

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

Arbitrary Matrix $3$

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

$\quad \mathbf A = \begin {pmatrix} 1 & 2 & 0 \\ 0 & -1 & 2 \\ -1 & 2 & 0 \\ \end {pmatrix}$

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

$\quad \mathbf A^{-1} = \begin {pmatrix} \dfrac 1 2 & 0 & -\dfrac 1 2 \\ \dfrac 1 4 & 0 & \dfrac 1 4 \\ \dfrac 1 8 & \dfrac 1 2 & \dfrac 1 8 \\ \end {pmatrix}$