Recipe

Build an elementary matrix (and its inverse) for a row operation

Use when: you're told one specific row operation and asked for the matrix E that performs it, or for E^{-1}, without row-reducing anything

Goal: the matrix E for that operation, checked by multiplying it into a real matrix, plus E^{-1} if the problem asks for it

Taught in Elementary matrices

1. 1

   Every elementary matrix comes from exactly one row operation: a swap of two rows, a scaling of one row by a nonzero constant, or adding a multiple of one row to another.

   • a swap of rows i and j — build E by swapping those two rows of I_n.
   • scaling row i by c — build E by scaling row i of I_n by c.
   • adding c times row k to row i — build E by doing that same thing to I_n.

   Why: the three operations produce three different-looking matrices, so misreading which one you were handed builds the wrong E from the very first line.

2. 2

   Start from the identity matrix of the right size and apply only the single operation you identified — every other row of I_n stays untouched.

   Example

   Suppose the operation is "add 5 times row 1 to row 3" in I_3. Rows 1 and 2 of I_3 are left alone; only row 3 changes, by adding 5 times row 1 to it: I_3=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}\;\xrightarrow{R_3\to R_3+5R_1}\;E=\begin{pmatrix}1&0&0\\0&1&0\\5&0&1\end{pmatrix}

   Why: an elementary matrix is defined as I_n with exactly one row operation applied — building anything else isn't an elementary matrix at all.

3. 3

   Pick any matrix A with the right number of rows, compute EA, and confirm the result is A with only the intended row changed.

   Example

   For A=\begin{pmatrix}1&2&3\\0&1&4\\2&0&1\end{pmatrix}, EA=\begin{pmatrix}1&0&0\\0&1&0\\5&0&1\end{pmatrix}\begin{pmatrix}1&2&3\\0&1&4\\2&0&1\end{pmatrix}=\begin{pmatrix}1&2&3\\0&1&4\\7&10&16\end{pmatrix} Rows 1 and 2 of A passed through unchanged; row 3 became (2,0,1)+5(1,2,3)=(7,10,16), exactly the intended operation and nothing else.

   Why: this is the defining property of an elementary matrix being checked directly on real numbers, not assumed from having built E correctly on paper.

4. 4

   Every elementary row operation is undone by another elementary row operation: a swap by the same swap again, a scale by c by a scale by \frac1c, and adding c times row k to row i by adding -c times row k to row i.

   Example

   The operation above added 5 times row 1 to row 3, so its inverse subtracts the same thing back off: apply R_3 \to R_3 - 5R_1 to I_3. E^{-1}=\begin{pmatrix}1&0&0\\0&1&0\\-5&0&1\end{pmatrix}

   Why: undoing a single row operation is itself a single row operation, so its matrix is built the same way Step 2 built E — from I_n, not from row-reducing [E\mid I].

5. 5

   Multiply E and E^{-1} together and confirm the product is the identity.

   Example

   EE^{-1}=\begin{pmatrix}1&0&0\\0&1&0\\5&0&1\end{pmatrix}\begin{pmatrix}1&0&0\\0&1&0\\-5&0&1\end{pmatrix}=\begin{pmatrix}1&0&0\\0&1&0\\5-5&0&1\end{pmatrix}=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}

   Why: this confirms E^{-1} really is the inverse, not just a plausible-looking guess at the undo operation.