Editing Invertible matrix (section)

=== Gaussian elimination ===
[[Gaussian elimination]] is a useful and easy way to compute the inverse of a matrix. To compute a matrix inverse using this method, an [[augmented matrix]] is first created with the left side being the matrix to invert and the right side being the [[identity matrix]]. Then, Gaussian elimination is used to convert the left side into the identity matrix, which causes the right side to become the inverse of the input matrix.

For example, take the following matrix: <math display=block>\mathbf{A} = \begin{pmatrix}-1 & \tfrac{3}{2} \\ 1  & -1\end{pmatrix}. </math> 

The first step to compute its inverse is to create the augmented matrix <math display=block>\left(\!\!\begin{array}{cc|cc}
-1 & \tfrac{3}{2} & 1 & 0 \\
 1 & -1 & 0 & 1
 \end{array}\!\!\right) .</math> 

Call the first row of this matrix <math>R_1</math> and the second row <math>R_2</math>. Then, add row 1 to row 2 <math>(R_1 + R_2 \to R_2).</math> This yields <math display=block>\left(\!\!\begin{array}{cc|cc}
-1 & \tfrac{3}{2} & 1 & 0 \\
 0 & \tfrac{1}{2} & 1 & 1
\end{array}\!\!\right).</math> 

Next, subtract row 2, multiplied by 3, from row 1 <math>(R_1 - 3\, R_2 \to R_1),</math> which yields <math display=block>\left(\!\!\begin{array}{cc|cc}
-1 & 0 & -2 & -3 \\
 0 & \tfrac{1}{2} & 1 & 1
\end{array}\!\!\right).</math>

Finally, multiply row 1 by −1 <math>(-R_1 \to R_1)</math> and row 2 by 2 <math>(2\, R_2 \to R_2).</math> This yields the identity matrix on the left side and the inverse matrix on the right:<math display=block>\left(\!\!\begin{array}{cc|cc}
1 & 0 & 2 & 3 \\
0 & 1 & 2 & 2
\end{array}\!\!\right).</math> 

Thus, <math display=block>\mathbf{A}^{-1} = \begin{pmatrix}
2 & 3 \\
2 & 2
\end{pmatrix}.</math>
It works because the process of Gaussian elimination can be viewed as a sequence of applying left matrix multiplication using elementary row operations using [[Elementary matrix|elementary matrices]] (<math>\mathbf E_n</math>), such as <math>\mathbf E_n \mathbf E_{n-1} \cdots \mathbf E_2 \mathbf E_1 \mathbf A = \mathbf I.</math>

Applying right-multiplication using <math>\mathbf A^{-1},</math> we get <math> \mathbf E_n \mathbf E_{n-1} \cdots \mathbf E_2 \mathbf E_1 \mathbf I = \mathbf I \mathbf A^{-1}.</math> And the right side <math>\mathbf I \mathbf A^{-1} = \mathbf A^{-1}, </math> which is the inverse we want.  

To obtain <math> \mathbf E_n \mathbf E_{n-1} \cdots \mathbf E_2 \mathbf E_1 \mathbf I,</math> we create the augumented matrix by combining {{math|'''A'''}} with {{math|'''I'''}} and applying [[Gaussian elimination]]. The two portions will be transformed using the same sequence of elementary row operations. When the left portion becomes {{math|'''I'''}}, the right portion applied the same elementary row operation sequence will become {{math|'''A'''{{sup|−1}}}}.