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I've been working on matrices lately. Currently, I am stuck on solving systems of linear equations using matrices. I've read the following article which has proved very helpful in understanding the basics of matrices intuitively: An Intuitive Guide to Linear algebra, however, one thing still confuses me. In the following illustrations, pic A picture

One thing still confuses me. If the "operations" matrix and the "input data" matrix are switched, then the output changes. Therefore, I conclude they matrices are not commutative, but in the following example:

[3 2 1 ] [ x ] [ 5 ]
|2 3 4 | | y | = | 31 |
[2 6 1 ] [ z ] [ 43 ]

But say I want to multiply both sides of the matrix by a certain matrix. [ 2 ]
[ 5 |
[ 1 ]

are both these operations valid?

this:

[ 2 ] [3 2 1 ][ x ] = [ 2 ] [ 5 ]
[ 5 ] |2 3 4 || y | = [ 5 ] | 31 |
[ 1 ] [2 6 1 ] [ z ] = [ 1 ] [ 43 ]

...and this:

[3 2 1 ] [ x ] [ 2 ] = [ 5 ] [ 2 ]
|2 3 4 | | y | [ 5 ]= | 31 | [ 5 ]
[2 6 1 ] [ z ] [ 1 ] = [ 43 ] [ 1 ]

Anyways, can anyone explain to me how I can simplify,

[3 2 1 ]
|2 3 4 |
[2 6 1 ]

into an identity matrix.

In other words, how can I simplify a system of linear equations?

Thanks.

Colin

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1 Answer 1

are both these operations valid

No. I think what you want to do is right-multiply both sides by a $3\times 3$ matrix whose diagonal elements are $2,5,1$.

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That didn't work. I tried it. –  Colin May 28 at 23:52

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