Two matrix A ,B which are Row equivalence

Question

I have two matrix $A,B$ which are row equivalence,I have to find a matrix $P$ such that $A=PB$ .

I try solving that by saying that $P$ is elementary procedure therefore I try find which elementary procedure i need to do for going from $B$ to $A$. I try to guess it, but i am sure there is a way to do so,and I do not know how.

I DUNNO WHY IT doesnt make space, $B, A \in \mathbb{R}^{ 3\times 2}$ .

I need to find $A=PB$ thank you

$$B= \begin{bmatrix} 1 & -3 \\ -4 & 1 \\ 6 & 9 \end{bmatrix}, A= \begin{bmatrix} -4 & 1 \\ -2 & -10 \\ 12 & 18 \end{bmatrix}$$

http://meta.math.stackexchange.com/a/5023/306553 mathjax help in typing matrices. — Siong Thye Goh, Dec 28 '16 at 18:13

Siong Thye Goh · Accepted Answer · 2016-12-28T20:36:13.293

0

Let's break down this into smaller problems:

It is known that the reduced row echelon form is $$R=\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{bmatrix}$$ Task $1$: find a nonsingular matrix $Q$ such that $QA=R$, record down the elementary operations that you use.

Task $2$: find a nonsingular matrix $W$ such that $WB=R$, record dow the elementary operations that you use.

Task $3$: Since $Q$ is invertible, $QA=R$ and $WB=R$, conclude that $QA=WB$ and $A=Q^{-1}W$.

Task $4$: Compute $P=Q^{-1}W$.

Edit:

How to find $W$ such that $WB=R$.

Let $E_1$ be the operation $4R_1+R_2$, i.e. multiply the first row by $4$ and add to row $2$. the second row become $[0, -11]$

$E_2$ be the operation $-\frac{1}{11}R_2$. The second row become $[0, 1]$

$E_3$ be the operation $3R_2+R_1$. The first row become $[1, 0]$.

$E_4$ be the operation $-6R_1+R_3$. The third row become $[0,9]$

$E_5$ be the operation $-9R_2+R_3$. The third row become $[0,0]$.

$E_5E_4E_3E_2E_1B=R$, here $W=E_5E_4E_3E_2E_1$.

Similarly $Q$ in $QA=R$ can be expressed as elementary operations, $Q=F_m \ldots F_1$.

$$Q^{-1}W=(F_m \ldots F_1)^{-1}(E_5 \ldots E_1)=F_1^{-1}\ldots F_m^{-1}E_5\ldots E_1$$

To compute $P$, it suffices to perform the operations $F_1^{-1}\ldots F_m^{-1}E_5\ldots E_1$ on the identity matrix.

edited Dec 28 '16 at 20:36

answered Dec 28 '16 at 18:10

Siong Thye Goh

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I dont understand,i do here with my example – Moshe Levy Dec 28 '16 at 18:16
are you able to find the matrix $Q$ and $W$? also in your question, did I understand correctly that $B=\begin{bmatrix} 1 &-3 \ -4 &1 \6 & 9 \end{bmatrix}$ – Siong Thye Goh Dec 28 '16 at 18:20
Yes this ,thanks – Moshe Levy Dec 28 '16 at 18:22
are you able to find $Q$ and $W$? – Siong Thye Goh Dec 28 '16 at 18:23
Reduced row encholn you mean canocial form? – Moshe Levy Dec 28 '16 at 18:29
I cant who is P here – Moshe Levy Dec 28 '16 at 18:31
$P=Q^{-1}W$, find $Q$, find its inverse, then find $W$, then you can compute $P$ – Siong Thye Goh Dec 28 '16 at 18:39
The condition in Task 3 is redundant. You’ve already stated in Task 1 that $Q$ is nonsingular. – amd Dec 28 '16 at 19:27
@MosheLevy Recall that if you augment a matrix with an appropriately-sized identity matrix and row-reduce, you end up with the matrix that transforms your original into the rref. That is, row-reducing $\begin{bmatrix}A\mid I_3\end{bmatrix}$ produces $\begin{bmatrix}R\mid Q\end{bmatrix}$. – amd Dec 28 '16 at 19:35
Yes thank you ,is there a faster way to do so? – Moshe Levy Dec 28 '16 at 19:38

Two matrix A ,B which are Row equivalence

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