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Factorising 3x3 Matrices

I've been having trouble working out how you factorise matrices.


The textbook says:





I know how to calculate the final factor of 'M' once it is 1 off being fully factorised (C1 . C2 x C3), but I can't seem to find any factors that are actually correct. The example matrix seems too simple to apply any things learnt to question 2, anyway.


Is it possible that somebody could write out a worked example to 2) so that I can learn some of the techniques used? I need to eventually learn how to factorise things like this:







Thanks.


(Can't ask people irl because this is self-taught :\)
Original post by fatart123
...


For 2)

We can see that "a" is a factor for column 1, and similiarly "b" for column 2 and "c" for column 3.

So we can take out these factors to get:

abc∣111abca2b2c2∣abc\begin{vmatrix} 1 & 1 & 1 \\a & b & c \\a^2 & b^2 & c^2 \end{vmatrix}

If we subtract column2 from column1 we get:

abc∣011a−bbca2−b2b2c2∣abc\begin{vmatrix} 0 & 1 & 1 \\a-b & b & c \\a^2-b^2 & b^2 & c^2 \end{vmatrix}

and we can see a factor "a-b" which we can take out to get:

abc(a−b)∣0111bca+bb2c2∣abc(a-b)\begin{vmatrix} 0 & 1 & 1 \\1 & b & c \\a+b & b^2 & c^2 \end{vmatrix}

We can now subtract column 3 from column 2 and take out a factor "b-c" to get:

abc(a−b)(b−c)∣00111ca+bb+cc2∣abc(a-b)(b-c)\begin{vmatrix} 0 & 0 & 1 \\1 & 1 & c \\a+b & b+c & c^2 \end{vmatrix}

And at this stage, I'd expand by row 1 to get:

abc(a−b)(b−c)(c−a)abc(a-b)(b-c)(c-a)
(edited 10 years ago)
Reply 2
Avatar for fatart123
fatart123
OP
15
Thanks a lot! That was definitely the hardest thing I've ever come across in maths, lol.
Original post by fatart123
Thanks a lot! That was definitely the hardest thing I've ever come across in maths, lol.


It comes easier with practise.
Reply 4
Avatar for fatart123
fatart123
OP
15
I hope so. I have two years to practise this, anyways :wink:
I know its been pretty long but, how do we expand by row 1?
Original post by Charybdia
I know its been pretty long but, how do we expand by row 1?


If you're still unsure, please make a new thread about this since it's an old thread.

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