How to find an elementary elimination matrix

When we row-reduce a matrix, each step changes one or two of its rows. We can write each step as a matrix multiplication. Let’s build the matrices for two operations and see how they fit together.

Learn Linear Algebra with meStep-by-step videos, lesson notes, and practice for matrices, vectors, and transformations.

Explore Linear Algebra →

Do the row operation to the identity matrix

Start with $I_3$. To multiply the first row of a three-row matrix by $-1$, multiply the first row of $I_3$ by $-1$. This gives $E_1$.

$E_1=\left[\begin{array}{rrr}-1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

In $E_{1}A$, the first row of $E_1$ takes $-1$ times the first row of $A$. The next two rows of $E_1$ pick out the second and third rows of $A$.

Build a second matrix for the next row operation

To replace row 2 with row 2 plus five times row 1, start from a fresh identity matrix. Its second row becomes $(5,1,0)$.

$E_2=\left[\begin{array}{rrr}1& 0& 0\\ 5& 1& 0\\ 0& 0& 1\end{array}\right]$

Multiply in the order the operations are applied

Applying $E_1$ first and then $E_2$ gives $E_2\left(E_{1}A\right)$. Group the first two matrices to get $\left(E_2{E}_1\right)A$. The combined matrix does both steps at once.

$E_2{E}_1=\left[\begin{array}{rrr}-1& 0& 0\\ -5& 1& 0\\ 0& 0& 1\end{array}\right]$

For a longer sequence, each new elementary matrix goes on the left.

Full Linear Algebra course

Learn Linear Algebra with me

We’ll work through matrices, vectors, and transformations, one step at a time.

Explore my Linear Algebra course →

• Video lessons
• Lesson notes
• Practice problems

Included with membership, along with every other Krista King Math course.