For the matrices given, find an invertible matrix $U$ such that $UA=B$ and express $U$ as a product of elementary matrices.

$$A= \left[\begin{array}{r}2&-1&0\\1&1&1\end{array}\right]$$ $$B= \left[\begin{array}{r}3&0&1\\2&-1&0\end{array}\right]$$

I've found the invertible matrix $U$ and have expressed it as a product of elementary matrices. I found that:

$$U= \left[\begin{array}{r}1&1\\1&0\end{array}\right]$$

But, the second half of the question is: Can $B$ be replaced by any other $2 \times 3$ matrix? Why or why not?

I'm quite stuck on the second half here.

• Welcome to MSE. Please use MathJax better. – José Carlos Santos Mar 1 '18 at 23:42
• Can $B$ be replaced by any other $2\times3$ matrix ... and have what be true? – Greg Martin Mar 2 '18 at 0:02
• I don't exactly know what has to be true, so that's kind of why I'm here.. I don't quite fully understand the question. – Kee Mar 2 '18 at 0:09

Think about it this way: When you left-multiply a matrix $M$ by a row vector, the result is a linear combination of the rows of $M$ with the elements of the row vector as the coefficients. When you left-multiply $A$ by a matrix, each row of the result is basically the product of a row vector and $A$. How many ways are there of expressing the rows of $B$ in your question as linear combinations of the rows of $A$?