Inverse Example
Gauss Example
Solve $A^{-1}$ given the following invertible matrix $A$
\[
A=\begin{pmatrix} 3 & -2 & 4 \\ 2 & -4 & 5 \\1 & 8 & 2\end{pmatrix}
\]
- Step 0
\[
\begin{split}\begin{pmatrix}3 & -2 & 4 \\ 2 & -4 & 5 \\1 & 8 & 2\end{pmatrix}
\begin{pmatrix}1 & 0 & 0 \\0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\end{split}
\]
-
Step 1, 2, 3
- $L3 \iff L1$
- $L2 \leftarrow L2 - 3 L_1$
- $L3 \leftarrow L2 - 2 L_1$
\[
\begin{split}
\begin{pmatrix}
1 & 8 & 2 \\
0 & -26 & -2 \\
0 & -20 & 1 \\
\end{pmatrix}
\begin{pmatrix}
0 & 0 & 1 \\
1 & 0 & -3 \\
0 & 1 & -2 \\
\end{pmatrix}
\end{split}
\]
- ... a lot of steps ...
\[
\begin{split}\begin{pmatrix}1 & 0 & 0 \\0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}
\begin{pmatrix}8/11 & -6/11 & -1/11 \\ -1/66 & -1/33 & 7/66 \\-10/33 & 13/33 & 4/33\end{pmatrix} = A^{-1}\end{split}
\]
The second matrix is $A^{-1}$.
Cofactors Example
Given the following invertible matrix $A$ since $det(A)=66$. Using the cofactor matrix, solve $A^{-1}$.
\[
A=\begin{pmatrix} 3 & -2 & 4 \\ 2 & -4 & 5 \\1 & 8 & 2\end{pmatrix}
\]
\[ C = \begin{pmatrix} + det(\begin{pmatrix}-4 & 5 \\ 8 & 2\end{pmatrix}) & - det(\begin{pmatrix}2 & 5 \\ 1 & 2\end{pmatrix}) & + det(\begin{pmatrix}2 & -4 \\ 1 & 8\end{pmatrix}) \\ - det(\begin{pmatrix}-2 & 4 \\ 8 & 2\end{pmatrix}) & + det(\begin{pmatrix}3 & 4 \\ 1 & 2\end{pmatrix}) & - det(\begin{pmatrix}3 & -2 \\ 1 & 8\end{pmatrix}) \\ + det(\begin{pmatrix}-2 & 4 \\ -4 & 5\end{pmatrix}) & - det(\begin{pmatrix}3 & 4 \\ 2 & 5\end{pmatrix}) & + det(\begin{pmatrix}3 & -2 \\ 2 & -4\end{pmatrix}) \end{pmatrix} \] \[ \Leftrightarrow C = \begin{pmatrix} -48 & 1 & 20 \\ 36 & 2 & -26 \\ 6 & -7 & -8 \\ \end{pmatrix} \] And \[ C^T = \begin{pmatrix} -48 & 36 & 6 \\ 1 & 2 & -7 \\ 20 & -26 & -8 \\ \end{pmatrix} \]Then we have
\[ \begin{split} A^{-1} = \frac{1}{-66} * \begin{pmatrix} -48 & 36 & 6 \\ 1 & 2 & -7 \\ 20 & -26 & -8 \\ \end{pmatrix} \\ \Leftrightarrow \begin{pmatrix} 48/66 & -36/66 & -6/66 \\ -1/66 & -2/66 & 7/66 \\ -20/66 & 26/66 & 8/66 \\ \end{pmatrix} \Leftrightarrow \begin{pmatrix} 8/11 & -6/11 & -1/11 \\ -1/66 & -1/33 & 7/66 \\ -10/33 & 13/33 & 4/33 \\ \end{pmatrix} \end{split} \]