# Given matrix $X$, how to find elementary matrices $E_1$, $E_2$ and $E_3$ such that $X = E_1 E_2 E_3$?

Given $$X = \begin{bmatrix} 0 & 1\\ -2 & -18\end{bmatrix}$$ find elementary matrices $$E_1$$, $$E_2$$ and $$E_3$$ such that $$X = E_1 E_2 E_3$$.

My attempt

I did 3 row operations from $$X$$ to get to $$I_2$$

1. Swapping row 1 and row 2

2. Row 1 becomes $$-\frac12$$ of row 1

3. Row 1 becomes Row 1 - 9 Row 2

So then

$$E_1 = \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}, \qquad E_2 = \begin{bmatrix} -1/2 & 0\\ 0 & 1 \end{bmatrix}, \qquad E_3 = \begin{bmatrix} 1 & -9\\ 0 & 1 \end{bmatrix}$$

However, when I multiply the $$E_1$$, $$E_2$$ and $$E_3$$ it doesn't give $$X$$. Can someone please tell me where I have made a mistake or if I've approached this question incorrectly?

From the row operations you've performed, we can say that $$E_3E_2E_1 X=I$$. So, $$X=E_1^{-1}E_2^{-1}E_3^{-1}$$.

• I multiplied 𝐸3𝐸2𝐸1𝑋 but it does not give me I. Can you tell if my elementary matrices are incorrect? Aug 26, 2021 at 6:44

To complement Umesh's answer, using SymPy:

>>> from sympy import *
>>> X = Matrix([[  0,   1],
[ -2, -18]])
>>> E1 = Matrix([[0, 1],
[1, 0]])
>>> E2 = Matrix([[-Rational(1,2), 0],
[             0, 1]])
>>> E3 = Matrix([[1, -9],
[0,  1]])
>>> E3 * E2 * E1 * X
Matrix([[1, 0],
[0, 1]])

• so I should just find the inverse of E1, E2 and E3 that I found previously and those would be my answer? Aug 26, 2021 at 8:10
• @Hayley Yes, and note that the inverses of elementary matrices are very easy to compute. Aug 26, 2021 at 8:15