If we have a matrices \[A=\begin{bmatrix}a & b \\c & d \end{bmatri

Clifton Sanchez

Clifton Sanchez

Answered question

2021-11-23

If we have a matrices
A=[abcd] and e12(λ)=[1λ01]
then by doing product Ae12(λ)=[aaλ+bccλ+d] and e12(λ)A=[a+cλb+dλcd]
we can interpret that right multiplication by e12 to A gives a column-operation: add λ-times first column to the second column. In similar way, left multiplication by e12(λ) to A gives row-operation on A.
Is there any conceptual (not computational, if any) way to see that elementary row and column operations on a matrix can be expressed as multiplication by elementary matrices on left or right, accordingly?

Answer & Explanation

Froldigh

Froldigh

Beginner2021-11-24Added 17 answers

Here conceptual and computational ideas go hand in hand. We can see this by looking at the multiplication of A with e12(λ) in some detail.
We have
[1λ01]=[1001]+[0λ00]=I+λ[0100]
It is the position {12} of the blue marked 1 which determines selected row resp. column of A.
We obtain
Ae12(λ)=A(I+λ[0100])=A+λ[0a0c]
e12(λ)A=(I+λ[0100])A=A+λ[cd00]
Louis Smith

Louis Smith

Beginner2021-11-25Added 14 answers

Let A,B be two matrices of order n.
We can describe B as B=(b1,,bn), where bi is its column i.
Notice that AB=(Ab1,,Abn)
Now apply some column-operation on AB.
For example, let's say that its column i is multiplied by λ
So we obtain (Ab1,,λAbi,,Abn)
Notice that (Ab1,,λAbi,,Abn)=A(b1,,λbi,,bn)
So in order to apply some column-operation on AB, we can first apply it on B and then multiply the resulting matrix with A.
Can you see what happens if B=Id?
The same reasoning can be used for row-opperations.

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