Calculating the Inverse of a Matrix

19ty84 · 2025-10-20 04:25:47 · 学习·文化课

Calculating the Inverse of a Matrix

Calculating the inverse of a 4 \times 4 matrix needs at most 224 steps. Wow.

Step 0

Let

\begin{pmatrix} 2 & 1 & 1 \\ -5 & -3 & -3 \\ 0 & 2 & 0 \end{pmatrix}

be the matrix you want to calculate.

Step 1: Create an Augmented Matrix

Add I to the right-hand side of the matrix. In this example, it becomes

\begin{pmatrix} 2 & 1 & 1 & 1 & 0 & 0\\ -5 & -3 & -3 & 0 & 1 & 0\\ 0 & 2 & 0 & 0 & 0 & 1 \end{pmatrix}

Step 2: Calculating the Reduced Row Echelon Form

I hate this step.

Anyway, you want to make the left-hand side of the matrix to be I. You can do it through row operations:

Swap 2 rows.
Multiply each element in one row by a non-zero constant.
Add a multiple of another row to one row.

If you don't know how to do that, there are a lot of online resources about it.

Finally, we get

\begin{pmatrix} 1 & 0 & 0 & 3 & 1 & 0\\ 0 & 1 & 0 & 0 & 0 & \dfrac{1}{2}\\ 0 & 0 & 1 & -5 & -2 & -\dfrac{1}{2} \end{pmatrix}

Step 3: The right-hand side of this matrix is the inverse of the original matrix.

So the inverse of the matrix is

\begin{pmatrix} 3 & 1 & 0 \\ 0 & 0 & \dfrac{1}{2} \\ -5 & -2 & -\dfrac{1}{2} \end{pmatrix}

Why does it work?

Doing a row operation is actually left-multiplying an elementary matrix. When the left-hand side becomes I, the combination of those elementary matrices is A^{-1} because A^{-1}A=I. Since the same operations are done to the right-hand side, it becomes A^{-1}I=A^{-1}.