Question

Can someone please help me with the following question I have to submit before tomorrow.

Suppose I is the 3x3 identity matrix and A and B are 3x3 matrices such that: E1E2A = B

Where:
E1is the elementary matrix obtained by performing -0.5 (Row 2) gives(=) Row 2 on I, and

E2 is the elementary matrix obtained by performing Row 1 + 0.5 Row 2 gives(=) Row 1 on I.

a) Write down E1 and E2
b) Write down E1 inverse and E2 inverse

Answer 1

not sure what you mean by "gives (=)"

Answer 2

\[\rightarrow\]

Answer 3

sorry ignore the word gives then...
Just =. or you may even see it as a right arrow.

Answer 4

ah
so all you need to do is take the identity matrix and perform the given row operation to find En

Answer 5

\[\textbf I=\left[\begin{array}\ 1&0&0 \\0&1&0\\0&0&1\end{array}\right]\]

Answer 6

I'm not completely catching on... I have I as above yes...

Answer 7

does E1 then =
\[\left[\begin{matrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1\end{matrix}\right]\]

Answer 8

your missing the negative

Answer 9

I see, Thus I must make the 2 a -2 in order for it to give poitive 1 when multiplied with -0.5?

Answer 10

\[\textbf I=\left[\begin{array}\ 1&0&0 \\0&1&0\\0&0&1\end{array}\right]\] \[\sim \left[\begin{array}\ 1&0&0 \\0&-\frac12&0\\0&0&1\end{array}\right] \qquad -\frac{R_2}2 \longrightarrow R_2\]

Answer 11

And E2 =
\[\left[\begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1\end{matrix}\right]\]

Answer 12

Is that the correct final matrix of what E1 should be? Bit confused now.

Answer 13

\[\textbf I=\left[\begin{array}\ 1&0&0 \\0&1&0\\0&0&1\end{array}\right]\] \[\sim \left[\begin{array}\ 1&\frac 32&0 \\0&1&0\\0&0&1\end{array}\right] \qquad {R_1}+\frac {R_2}2 \longrightarrow R_1\]

Answer 14

I assume that 3/2 in position 1,2 should be 1/2

Answer 15

Yes because I find the problem there...
Let me check

Answer 16

Phi do you feel that this is the correct answer as to what matrix E2 should be?

Answer 17

That row operations according to me doesn't give R1 on I

Answer 18

the E2 matrix is I with the 2,1 position replaced with 1/2. If you multiply E2 times any 3x3 matrix it will add 1/2 row2 to row1 and put the answer in row 1
the other 2 rows of the matrix will be copied over.

when they say
E2 is the elementary matrix obtained by performing Row 1 + 0.5 Row 2 gives(=) Row 1 on I.
they must mean when E2 is applied to matrix A ?

Answer 19

oh yeah 0 isn't 1 my mistake

Answer 20

Thus E1 =
\[\left[\begin{matrix}1 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0& 1\end{matrix}\right]\]

Answer 21

And E2=

Answer 22

you mean -1/2 for E1 and -2 for the inverse of E1, right?

Answer 23

\[\left[\begin{matrix}1 & 0.5 & 0 \\ 0 & 1 & 0\\ 0 & 0& 1\end{matrix}\right]\]

Answer 24

Sorry looks like I still miss something apparently...:(

Answer 25

that is E2. For the inverse of E2, negate the "key entry" (the 0.5)

Answer 26

E1is the elementary matrix obtained by performing -0.5 (Row 2)
that is -0.5 not -2

Answer 27

-0.5 x (Row 2 of which matrix) to give what value?

Answer 28

Here is an example of what is going on.

Answer 29

Is there more to this question? Do they want you to find an A and B ?

Answer 30

Yes :
Write A as a product of E1 inverse, E2 inverse and B.
Then determine A if B =
\[\left[\begin{matrix}3 & 2 & 1 \\ -2 & 0 & 1 \\ 3 &1 & 1\end{matrix}\right]\]

Answer 31

OK, everything makes sense except where they say "gives(=) Row 2 on I "

the elementary matrix is defined by the row operations (not by the results)

but you have E1 and E2 and can check that they do the specified row operation.

with B= E1*E2*A
then A= E2i*E1i*B (E1i is E1 inverse)

Answer 32

When they ask me to Write A as a product of E1 inverse, E2 inverse and B. Must I multiply all 3 these matrices to obtain 1 matrix (the product)?

Answer 33

I assume they want what I just posted, and then use that "formula" by plugging in the known B to find A

Answer 34

Thank you so much, I'm completing the question now, if I wonder about anything still, I will ask. Thank you for all the help!

Answer 35

yw

Answer 36

phi, I get the matrix A as \[\left[\begin{matrix}4 & 2 & 0.5 \\ 4 & 0 & -2\\ 3 & 1 & 1\end{matrix}\right]\] is this correct?

Answer 37

No, It looks like you applied the inv(E)'s in the wrong order.
You were told
\[ E_1E_2A = B \]
left multiply both sides by inverse(E1):
\[ E_1^{-1}E_1E_2A = E_1^{-1}B \]
the left side simplifies to
\[ E_2A = E_1^{-1}B \]
now left multiply by the inverse of E2
\[E_2^{-1} E_2A = E_2^{-1} E_1^{-1}B \]
or
\[A = E_2^{-1} E_1^{-1}B \]

as a check, you should get B back using
\[ B=E_1E_2A \]