Question

what is a 3x3 invertible matrix?

Answer 1

Are you OK with what a 3x3 matrix is?

Answer 2

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

Answer 3

a solved example would be great

Answer 4

ok

Answer 5

@mathsmind ???

Answer 6

which means its determinant cant be 0?

Answer 7

yest i will show you the full steps

Answer 8

consider the following matrix

Answer 9

PLZ continue

Answer 10

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

Answer 11

Step one:
\[A^T\]

Answer 12

which means transpose of rows and columns of A

Answer 13

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

Answer 14

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

Answer 15

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

Answer 16

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

Answer 17

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

Answer 18

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

Answer 19

so we need to find adjA

Answer 20

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

Answer 21

ok sure

Answer 22

any idea?

Answer 23

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

Answer 24

i will show you shortly

Answer 25

@mathsmind ???

Answer 26

@mathsmind why is taking so long?

Answer 27

sorry back, am at work

Answer 28

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

Answer 29

yep

Answer 30

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

Answer 31

But it would be really long

Answer 32

well this is the world of Matrices...

Answer 33

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

Answer 34

or subtracting p^4-2P=0 in order to find the new P?

Answer 35

Isn't there any short way

Answer 36

have you done hermitian matrices...

Answer 37

nope

Answer 38

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

Answer 39

okay

Answer 40

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

Answer 41

see ya tonight

Answer 42

@ParthKohli

Answer 43

lol, you still here? I thought you got help

Answer 44

No

Answer 45

can u do it?

Answer 46

Short way

Answer 47

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

Answer 48

Nope.

Answer 49

http://www.wikihow.com/Inverse-a-3X3-Matrix

Answer 50

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