How Do Elementary Matrices Determine Matrix Multiplication Order?

[brushman]

Homework Statement

Which three matrices E₂₁, E₃₁, E₃₂, put A into triangular form U?

A =
[1 1 0]
[4 6 1]
[-2 2 0]

and E₃₂E₃₁E₂₁A = U.

Multiply those E's to get one matrix M that does elimination: MA = U.

Attempt at a solution

I already found all the E's. What I don't get is, if it didn't tell me E₃₂E₃₁E₂₁A = U, how would I know the correct order of multiplication in for M? In other words, how do I know M = E₃₂E₃₁E₂₁ instead of E₂₁E₃₁E₃₂

edit- Thanks Tedjn, HallsofIvy

 

[Tedjn]

The job of matrix E_mn is to place a zero in the m,n position of the matrix. Following the usual row reduction algorithm, it is important that we do the entire first column first, then the second column, etc. (Why is this order important?) The closest matrix to A is E₂₁ because our first step is to place a zero in the 2,1 position, and we perform that multiplication first.

 

[HallsofIvy]

The simplest way to put a matrix into triangular form is to "row reduce" it. And each "row operation" corresponds to an elementary matrix- the matrix you get by applying that row operation to the identity matrix. Applying a row operation to a matrix is the same as multiplying the matrix by the corresponding elementary matrix- and a series of elementary matrices is multiplied in the same order the row operations were applied.

For example, here, the first row operation I would use to reduce A to diagonal form would be to subtract four times the first row from the second: that corresponds to elementary matrix
$$\begin{bmatrix}1 & 0 & 0 \\ -4 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$
and is done by the matrix multiplication
$$\begin{bmatrix}1 & 0 & 0 \\ -4 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\begin{bmatrix}1 & 1 & 0 \\ 4 & 6 & 1 \\- 2 & 2 & 0\end{bmatrix}$$

The second thing I would do would be to add two times the first row to the third row:
$$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 2 & 0 & 1\end{bmatrix}\begin{bmatrix}1 & 0 & 0 \\ -4 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\begin{bmatrix}1 & 1 & 0 \\ 4 & 6 & 1 \\- 2 & 2 & 0\end{bmatrix}$$

It will take a total of 6 row reductions (two per column) to reduce a 3 by 3 matrix to a diagonal matrix. Since you are asked for three matrices, I suspect they intend for you to multiply the two matrices corresponding to a single column together. That is, the first matrix on the right for your matrix multiplication would be
$$\begin{bmatrix}1 & 0 & 0 \\ -4 & 1 & 0 \\ 2 & 0 & 1\end{bmatrix}$$