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?

Question

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?

Froldigh · Accepted Answer

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 · Accepted Answer

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&#039;s say that its column i is multiplied by λSo 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.

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

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