LOOSEr
what is a 3x3 invertible matrix?



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tkhunny
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Are you OK with what a 3x3 matrix is?

mathsmind
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do you want a formal definition or a solved example showing all steps of inverting a 3x3 matrix?

LOOSEr
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a solved example would be great

mathsmind
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ok

LOOSEr
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@mathsmind ???

LOOSEr
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which means its determinant cant be 0?

mathsmind
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yest i will show you the full steps

mathsmind
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consider the following matrix

LOOSEr
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PLZ continue

mathsmind
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\[A=\left[\begin{matrix}1 & 2 & 0\\ 3& 1 & 5\\ 1 & 2 & 3\end{matrix}\right]\]

mathsmind
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Step one:
\[A^T\]

mathsmind
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which means transpose of rows and columns of A

mathsmind
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\[A^T=\left[\begin{matrix}1 & 3 & 1\\ 2& 1 & 2\\ 0 & 5 & 3\end{matrix}\right]\]

mathsmind
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Step Two: find the adjoint or adjugate of A, which implies that we replace each element by their cofactor...

LOOSEr
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If P = \begin{bmatrix}
a &b &c \\
d &e &f \\
g &h &i
\end{bmatrix}

mathsmind
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now please concentrate on this step it is simple but requires a bit of accuracy

LOOSEr
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And P^4=2P
how can we find P

mathsmind
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let's finish this first part then i will come to the 2nd question, you asked me first for a numerical example

mathsmind
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so we need to find adjA

LOOSEr
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I know how to find a inverse matrix.
So, can u plz do the 2nd question

mathsmind
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ok sure

LOOSEr
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any idea?

mathsmind
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yes sorry am on the phone, but just a quick hint for P^n we use the principle of induction

mathsmind
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i will show you shortly

LOOSEr
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@mathsmind ???

LOOSEr
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@mathsmind why is taking so long?

mathsmind
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sorry back, am at work

mathsmind
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now P^4 means you are multiplying your matrix by itself 4 times.

LOOSEr
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yep

mathsmind
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you know how to multiply the, it would be PP=P^2, then P^2P=P^3. then P^3P=P^4

LOOSEr
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But it would be really long

mathsmind
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well this is the world of Matrices...

mathsmind
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are you thinking of algebraic operation on matrices such as taking the inverse of 2P

mathsmind
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or subtracting p^42P=0 in order to find the new P?

LOOSEr
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Isn't there any short way

mathsmind
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have you done hermitian matrices...

LOOSEr
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nope

mathsmind
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listen can we kindly finish this tonight because it is 5 am and i need to go ...

LOOSEr
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okay

mathsmind
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i will provide you with full solution even if you are not online ok

mathsmind
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see ya tonight

LOOSEr
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@ParthKohli

Hero
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lol, you still here? I thought you got help

LOOSEr
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No

LOOSEr
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can u do it?

LOOSEr
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Short way

ParthKohli
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I haven't done matrices yet: I am stuck with number theory at the moment.

ParthKohli
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Nope.


sauravshakya
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@looser there are many matrices such that P^4=2P
but only one is invertible which is dw:1361026643638:dw