[Math] Finding the inverse of a matrix using elementary matricies

inverselinear algebramatrices

Can somebody help me understand what exactly is being asked here? I understand how to construct elementary matrices from these row operations, but I'm unsure what the end goal is. Am I to assume that $Y$ is built from these row operations?

Let $Y$ be the $4\times 4$ matrix which applies the following operations, in the order given, when used as a left-multiplier:

  1. divide $R_{2}$ by $3$,
  2. add $2R_{1}$ to $R_{4}$,
  3. swap $R_{2}$ and $R_{4}$, then
  4. subtract $R_{4}$ from $R_{3}$.

Find $Y^{-1}$ without calculating $Y$.

If I were to venture a guess, I would say that it's implying that $E_{4}E_{3}E_{2}E_{1}=Y$, and therefore I need to find $Y^{-1}=E^{-1}_{1}E^{-1}_{2}E^{-1}_{3}E^{-1}_{4}$. But the wording of the question makes me not 100% sure.

I've found \begin{align*}
E_{4}E_{3}E_{2}E_{1}=\left[\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & -1 \\
0 & 0 & 0 & 1
\end{array}\right]\left[\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0
\end{array}\right]\left[\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
2 & 0 & 0 & 1
\end{array}\right]\left[\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & \frac{1}{3} & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right]
\end{align*}

so if my assumption is right, I'd just need to build the inverse of the matrices and multiply them in reverse order to get $Y^{-1}$

Best Answer

Yes, that is correct. The inverse of an elementary operation matrix is the operation that reverses that elementary operation:

  1. Add $R_3$ to $R_4$
  2. Swap $R_2$ and $R_4$
  3. Subtract $2R_1$ from $R_4$
  4. Multiply $R_2$ by $3$

Apply these transformations to the identity matrix to find the inverse of $Y$.

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