Calculating Inverse Matrices

There are two main methods to obtain the Inverse Matrix. Remember that onlyr[r(Matrix%20Variants.md#Square|square%20matrices) can have inverses.

Via Determinant §

This method is much more complicated for square matrices with more than two rows and columns, but can be calculated:

$${\mathbf{A}}^{-1}=\frac{1}{\det (\mathbf{A})}\text{adj}(\mathbf{A})$$

where $\text{adj}(\mathbf{A})$ is the Adjugate Matrix#todo and $\det (\mathbf{A})$ is the Determinant of $\mathbf{A}$r

In a $2\times 2$ matrix:

$${\mathbf{A}}^{-1}=\frac{1}{\det (\mathbf{A})}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$$

Via Gaussian-Jordan Elimination §

We can obtain the inverse of a matrix by creating an augmented matrix containing the matrix $\mathbf{A}$ and the Identity Matrix, $\mathbf{I}$:

$$[\mathbf{A}|\mathbf{I}]$$

and use the Gaussian-Jordan elimination method to obtain the RREF of $\mathbf{A}$:

$$[\mathbf{A}|\mathbf{I}]\sim [\text{rref}(\mathbf{A})|{\mathbf{A}}^{-1}]$$

• If $\text{rref}(\mathbf{A})=\mathbf{I}$, then $\mathbf{A}$ is invertible, and has an inverse matrix
• If $\text{rref}(\mathbf{A})\ne \mathbf{I}$, then $\mathbf{A}$ is singular.

Examples §

Using RREF

Calculate the inverse matrix of the following matrix:

\boldsymbol{A} = \begin{bmatrix}

2 & 1 & 1 \ 3 & 2 & 1 \ 2 & 1 & 2 \end{bmatrix}

>\ >We start by appending the identity the matrix: >$$ >[\boldsymbol{A|I}] = \left[ \begin{array}{rrr|rrr} 2 & 1 & 1 & 1 & 0 & 0 \\ 3 & 2 & 1 & 0 & 1 & 0 \\ 2 & 1 & 2 & 0 & 0 & 1 \end{array} \right]

And then row reduce this:

[\boldsymbol{A|I}] = \left[ \begin{array}{rrr|rrr}

2 & 1 & 1 & 1 & 0 & 0 \ 3 & 2 & 1 & 0 & 1 & 0 \ 2 & 1 & 2 & 0 & 0 & 1 \end{array} \right] ~

#TODO