# Find an invertible matrix U such that UA = R is in reduced row-echelon form , and express U as a product of elementary matrices.

Find an invertible matrix $U$ such that $UA = R$ is in reduced row-echelon form , and express $U$ as a product of elementary matrices.

I have found R(RREF of $A$), but cannot do $U = R * A$ (cannot multiply 4x3 on 4x3)

• $E_i(c)$ to denote the elementary operation of “multiplying the $i$-th row by $c$” (for $c\ne0$);
• $E_{ij}(d)$ to denote the elementary operation of “adding to the $i$-th row the $j$-th row multiplied by $d$” (for $i\ne j$);
• $E_{ij}$ to denote the elementary operation of “switching the $i$-th and $j$-th rows (for $i\ne j$).
The operations you have to perform on your matrix to get the RREF are, in order, $$E_1(1/2)\quad E_{21}(-3)\quad E_{31}(-1)\quad E_{2}(-2/5)\quad E_{32}(5/2)\quad E_{12}(-1/2)$$ and the RREF form is $$R=\begin{bmatrix} 1 & 0 & 7/5 & 1/5 \\ 0 & 1 & -7/5 & -2/5 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ Thus we know that $$R= E_{12}(-1/2) E_{32}(5/2) E_{2}(-2/5) E_{31}(-1) E_{21}(-3) E_1(1/2) A$$