Matrix as a product of elementary matrices?

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So



$$A = beginbmatrix 1 & 2 \ 3 & 1 endbmatrix$$



and the matrix can be reduced in these steps:



$$beginbmatrix 1 & 2 \ 0 & -5 endbmatrix$$



via an elementary matrix that looks like this:



$$ E_1 = beginbmatrix 1 & 0 \ -3 & 1 endbmatrix$$



next:



$$left [ beginmatrix 1 & 0 \ 0 & -5 endmatrix right ] $$



via an elementary matrix that looks like this:



$$ E_2 = left [ beginmatrix 1 & frac25 \ 0 & 1 endmatrix right ] $$



next:



$$left [ beginmatrix 1 & 0 \ 0 & 1 endmatrix right ] $$



via an elementary matrix that looks like this:



$$ E_1 = left [ beginmatrix 1 & 0 \ 0 & frac-15 endmatrix right ] $$



So...



$$E_1^-1 = left [ beginmatrix 1 & 0 \ 3 & 1 endmatrix right ] $$
$$E_2^-1 = left [ beginmatrix 1 & frac-25 \ 0 & 1 endmatrix right ] $$
$$E_3^-1 = left [ beginmatrix 1 & 0 \ 0 & -5 endmatrix right ] $$



And if I multiply them together I get:



$$E_1^-1 * E_2^-2 = left [ beginmatrix 1 & frac-25 \ 3 & frac-15 endmatrix right ] = C$$



and



$$C * E_3^-1 = left [ beginmatrix 1 & 2 \ 3 & 1 endmatrix right ] $$



So this works out. Is this right?



Also question, the underlying premise of all of this is that $A$ is invertible right? And if $A$ is invertible, that means that a series of row operations can change it to the identity matrix. Why is this? This doesn't make intuitive sense to me.



Also, why does the product of elementary matrices equal $A$? What is the underlying theorem?




linear-algebra matrices gaussian-elimination




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  • Multiplying a matrix $A$ by an elementary matrix $E$ is equivalent to applying elementary row operations to $A$. The underlying theorem that you seek says that $A$ is invertible if and only if it is row-equivalent and column equivalent to the identity matrix; it is commonly called "the invertible matrix theorem". You can search up a proof in any elementary linear algebra textbook.
    – amarney
    Aug 6 at 19:18














up vote
0
down vote

favorite













enter image description here




So



$$A = beginbmatrix 1 & 2 \ 3 & 1 endbmatrix$$



and the matrix can be reduced in these steps:



$$beginbmatrix 1 & 2 \ 0 & -5 endbmatrix$$



via an elementary matrix that looks like this:



$$ E_1 = beginbmatrix 1 & 0 \ -3 & 1 endbmatrix$$



next:



$$left [ beginmatrix 1 & 0 \ 0 & -5 endmatrix right ] $$



via an elementary matrix that looks like this:



$$ E_2 = left [ beginmatrix 1 & frac25 \ 0 & 1 endmatrix right ] $$



next:



$$left [ beginmatrix 1 & 0 \ 0 & 1 endmatrix right ] $$



via an elementary matrix that looks like this:



$$ E_1 = left [ beginmatrix 1 & 0 \ 0 & frac-15 endmatrix right ] $$



So...



$$E_1^-1 = left [ beginmatrix 1 & 0 \ 3 & 1 endmatrix right ] $$
$$E_2^-1 = left [ beginmatrix 1 & frac-25 \ 0 & 1 endmatrix right ] $$
$$E_3^-1 = left [ beginmatrix 1 & 0 \ 0 & -5 endmatrix right ] $$



And if I multiply them together I get:



$$E_1^-1 * E_2^-2 = left [ beginmatrix 1 & frac-25 \ 3 & frac-15 endmatrix right ] = C$$



and



$$C * E_3^-1 = left [ beginmatrix 1 & 2 \ 3 & 1 endmatrix right ] $$



So this works out. Is this right?



Also question, the underlying premise of all of this is that $A$ is invertible right? And if $A$ is invertible, that means that a series of row operations can change it to the identity matrix. Why is this? This doesn't make intuitive sense to me.



Also, why does the product of elementary matrices equal $A$? What is the underlying theorem?




linear-algebra matrices gaussian-elimination




share|cite|improve this question





















  • Multiplying a matrix $A$ by an elementary matrix $E$ is equivalent to applying elementary row operations to $A$. The underlying theorem that you seek says that $A$ is invertible if and only if it is row-equivalent and column equivalent to the identity matrix; it is commonly called "the invertible matrix theorem". You can search up a proof in any elementary linear algebra textbook.
    – amarney
    Aug 6 at 19:18












up vote
0
down vote

favorite









up vote
0
down vote

favorite












enter image description here




So



$$A = beginbmatrix 1 & 2 \ 3 & 1 endbmatrix$$



and the matrix can be reduced in these steps:



$$beginbmatrix 1 & 2 \ 0 & -5 endbmatrix$$



via an elementary matrix that looks like this:



$$ E_1 = beginbmatrix 1 & 0 \ -3 & 1 endbmatrix$$



next:



$$left [ beginmatrix 1 & 0 \ 0 & -5 endmatrix right ] $$



via an elementary matrix that looks like this:



$$ E_2 = left [ beginmatrix 1 & frac25 \ 0 & 1 endmatrix right ] $$



next:



$$left [ beginmatrix 1 & 0 \ 0 & 1 endmatrix right ] $$



via an elementary matrix that looks like this:



$$ E_1 = left [ beginmatrix 1 & 0 \ 0 & frac-15 endmatrix right ] $$



So...



$$E_1^-1 = left [ beginmatrix 1 & 0 \ 3 & 1 endmatrix right ] $$
$$E_2^-1 = left [ beginmatrix 1 & frac-25 \ 0 & 1 endmatrix right ] $$
$$E_3^-1 = left [ beginmatrix 1 & 0 \ 0 & -5 endmatrix right ] $$



And if I multiply them together I get:



$$E_1^-1 * E_2^-2 = left [ beginmatrix 1 & frac-25 \ 3 & frac-15 endmatrix right ] = C$$



and



$$C * E_3^-1 = left [ beginmatrix 1 & 2 \ 3 & 1 endmatrix right ] $$



So this works out. Is this right?



Also question, the underlying premise of all of this is that $A$ is invertible right? And if $A$ is invertible, that means that a series of row operations can change it to the identity matrix. Why is this? This doesn't make intuitive sense to me.



Also, why does the product of elementary matrices equal $A$? What is the underlying theorem?




linear-algebra matrices gaussian-elimination




share|cite|improve this question














enter image description here




So



$$A = beginbmatrix 1 & 2 \ 3 & 1 endbmatrix$$



and the matrix can be reduced in these steps:



$$beginbmatrix 1 & 2 \ 0 & -5 endbmatrix$$



via an elementary matrix that looks like this:



$$ E_1 = beginbmatrix 1 & 0 \ -3 & 1 endbmatrix$$



next:



$$left [ beginmatrix 1 & 0 \ 0 & -5 endmatrix right ] $$



via an elementary matrix that looks like this:



$$ E_2 = left [ beginmatrix 1 & frac25 \ 0 & 1 endmatrix right ] $$



next:



$$left [ beginmatrix 1 & 0 \ 0 & 1 endmatrix right ] $$



via an elementary matrix that looks like this:



$$ E_1 = left [ beginmatrix 1 & 0 \ 0 & frac-15 endmatrix right ] $$



So...



$$E_1^-1 = left [ beginmatrix 1 & 0 \ 3 & 1 endmatrix right ] $$
$$E_2^-1 = left [ beginmatrix 1 & frac-25 \ 0 & 1 endmatrix right ] $$
$$E_3^-1 = left [ beginmatrix 1 & 0 \ 0 & -5 endmatrix right ] $$



And if I multiply them together I get:



$$E_1^-1 * E_2^-2 = left [ beginmatrix 1 & frac-25 \ 3 & frac-15 endmatrix right ] = C$$



and



$$C * E_3^-1 = left [ beginmatrix 1 & 2 \ 3 & 1 endmatrix right ] $$



So this works out. Is this right?



Also question, the underlying premise of all of this is that $A$ is invertible right? And if $A$ is invertible, that means that a series of row operations can change it to the identity matrix. Why is this? This doesn't make intuitive sense to me.



Also, why does the product of elementary matrices equal $A$? What is the underlying theorem?





linear-algebra matrices gaussian-elimination





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 6 at 21:41









Kyle

1,094617




1,094617









asked Aug 6 at 17:41









Jwan622

1,61211224




1,61211224











  • Multiplying a matrix $A$ by an elementary matrix $E$ is equivalent to applying elementary row operations to $A$. The underlying theorem that you seek says that $A$ is invertible if and only if it is row-equivalent and column equivalent to the identity matrix; it is commonly called "the invertible matrix theorem". You can search up a proof in any elementary linear algebra textbook.
    – amarney
    Aug 6 at 19:18
















  • Multiplying a matrix $A$ by an elementary matrix $E$ is equivalent to applying elementary row operations to $A$. The underlying theorem that you seek says that $A$ is invertible if and only if it is row-equivalent and column equivalent to the identity matrix; it is commonly called "the invertible matrix theorem". You can search up a proof in any elementary linear algebra textbook.
    – amarney
    Aug 6 at 19:18















Multiplying a matrix $A$ by an elementary matrix $E$ is equivalent to applying elementary row operations to $A$. The underlying theorem that you seek says that $A$ is invertible if and only if it is row-equivalent and column equivalent to the identity matrix; it is commonly called "the invertible matrix theorem". You can search up a proof in any elementary linear algebra textbook.
– amarney
Aug 6 at 19:18




Multiplying a matrix $A$ by an elementary matrix $E$ is equivalent to applying elementary row operations to $A$. The underlying theorem that you seek says that $A$ is invertible if and only if it is row-equivalent and column equivalent to the identity matrix; it is commonly called "the invertible matrix theorem". You can search up a proof in any elementary linear algebra textbook.
– amarney
Aug 6 at 19:18










1 Answer
1






active

oldest

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up vote
0
down vote













Given a matrix, the steps involved in determining a sequence of elementary matrices which, when multiplied together, give the original matrix is the same work involved in performing row reduction on the matrix.



For example, in your case you have
begingather
E_1 =
beginbmatrix
1 & 0 \
-3 & 1
endbmatrix
endgather



This is equivalent to performing the operation $R_2 -3R_1$ on $I_2$, the 2x2 identity matrix.
This row operation would be the first operation performed to reduce this matrix.
Personally, when I have to compute the sequence of elementary matrices I always do the reduction first and then compute the corresponding matrices.



Now, knowing that these are equivalent, we can clearly see why there is a relation to invertibility.
We know that if a matrix is invertible, it can be reduced to the identity matrix.
If it can be reduced to the identity matrix, then this means that there is a sequence of row operations involved in this reduction.
This in turn means that there is a corresponding sequence of elementary matrices of which the product forms the original matrix.






share|cite|improve this answer





















  • What do you mean by equivalent? Are the matrices equal?
    – Jwan622
    Aug 7 at 15:59










  • Yes -- by equivalent I meant that you end up with the same matrix, it's just two different ways of expressing the same work.
    – emarteca
    Aug 7 at 17:56










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active

oldest

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active

oldest

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active

oldest

votes








up vote
0
down vote













Given a matrix, the steps involved in determining a sequence of elementary matrices which, when multiplied together, give the original matrix is the same work involved in performing row reduction on the matrix.



For example, in your case you have
begingather
E_1 =
beginbmatrix
1 & 0 \
-3 & 1
endbmatrix
endgather



This is equivalent to performing the operation $R_2 -3R_1$ on $I_2$, the 2x2 identity matrix.
This row operation would be the first operation performed to reduce this matrix.
Personally, when I have to compute the sequence of elementary matrices I always do the reduction first and then compute the corresponding matrices.



Now, knowing that these are equivalent, we can clearly see why there is a relation to invertibility.
We know that if a matrix is invertible, it can be reduced to the identity matrix.
If it can be reduced to the identity matrix, then this means that there is a sequence of row operations involved in this reduction.
This in turn means that there is a corresponding sequence of elementary matrices of which the product forms the original matrix.






share|cite|improve this answer





















  • What do you mean by equivalent? Are the matrices equal?
    – Jwan622
    Aug 7 at 15:59










  • Yes -- by equivalent I meant that you end up with the same matrix, it's just two different ways of expressing the same work.
    – emarteca
    Aug 7 at 17:56














up vote
0
down vote













Given a matrix, the steps involved in determining a sequence of elementary matrices which, when multiplied together, give the original matrix is the same work involved in performing row reduction on the matrix.



For example, in your case you have
begingather
E_1 =
beginbmatrix
1 & 0 \
-3 & 1
endbmatrix
endgather



This is equivalent to performing the operation $R_2 -3R_1$ on $I_2$, the 2x2 identity matrix.
This row operation would be the first operation performed to reduce this matrix.
Personally, when I have to compute the sequence of elementary matrices I always do the reduction first and then compute the corresponding matrices.



Now, knowing that these are equivalent, we can clearly see why there is a relation to invertibility.
We know that if a matrix is invertible, it can be reduced to the identity matrix.
If it can be reduced to the identity matrix, then this means that there is a sequence of row operations involved in this reduction.
This in turn means that there is a corresponding sequence of elementary matrices of which the product forms the original matrix.






share|cite|improve this answer





















  • What do you mean by equivalent? Are the matrices equal?
    – Jwan622
    Aug 7 at 15:59










  • Yes -- by equivalent I meant that you end up with the same matrix, it's just two different ways of expressing the same work.
    – emarteca
    Aug 7 at 17:56












up vote
0
down vote










up vote
0
down vote









Given a matrix, the steps involved in determining a sequence of elementary matrices which, when multiplied together, give the original matrix is the same work involved in performing row reduction on the matrix.



For example, in your case you have
begingather
E_1 =
beginbmatrix
1 & 0 \
-3 & 1
endbmatrix
endgather



This is equivalent to performing the operation $R_2 -3R_1$ on $I_2$, the 2x2 identity matrix.
This row operation would be the first operation performed to reduce this matrix.
Personally, when I have to compute the sequence of elementary matrices I always do the reduction first and then compute the corresponding matrices.



Now, knowing that these are equivalent, we can clearly see why there is a relation to invertibility.
We know that if a matrix is invertible, it can be reduced to the identity matrix.
If it can be reduced to the identity matrix, then this means that there is a sequence of row operations involved in this reduction.
This in turn means that there is a corresponding sequence of elementary matrices of which the product forms the original matrix.






share|cite|improve this answer













Given a matrix, the steps involved in determining a sequence of elementary matrices which, when multiplied together, give the original matrix is the same work involved in performing row reduction on the matrix.



For example, in your case you have
begingather
E_1 =
beginbmatrix
1 & 0 \
-3 & 1
endbmatrix
endgather



This is equivalent to performing the operation $R_2 -3R_1$ on $I_2$, the 2x2 identity matrix.
This row operation would be the first operation performed to reduce this matrix.
Personally, when I have to compute the sequence of elementary matrices I always do the reduction first and then compute the corresponding matrices.



Now, knowing that these are equivalent, we can clearly see why there is a relation to invertibility.
We know that if a matrix is invertible, it can be reduced to the identity matrix.
If it can be reduced to the identity matrix, then this means that there is a sequence of row operations involved in this reduction.
This in turn means that there is a corresponding sequence of elementary matrices of which the product forms the original matrix.







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Aug 7 at 3:09









emarteca

1




1











  • What do you mean by equivalent? Are the matrices equal?
    – Jwan622
    Aug 7 at 15:59










  • Yes -- by equivalent I meant that you end up with the same matrix, it's just two different ways of expressing the same work.
    – emarteca
    Aug 7 at 17:56
















  • What do you mean by equivalent? Are the matrices equal?
    – Jwan622
    Aug 7 at 15:59










  • Yes -- by equivalent I meant that you end up with the same matrix, it's just two different ways of expressing the same work.
    – emarteca
    Aug 7 at 17:56















What do you mean by equivalent? Are the matrices equal?
– Jwan622
Aug 7 at 15:59




What do you mean by equivalent? Are the matrices equal?
– Jwan622
Aug 7 at 15:59












Yes -- by equivalent I meant that you end up with the same matrix, it's just two different ways of expressing the same work.
– emarteca
Aug 7 at 17:56




Yes -- by equivalent I meant that you end up with the same matrix, it's just two different ways of expressing the same work.
– emarteca
Aug 7 at 17:56












 

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