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Halsafar

Multiplicative-Identity Property

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What exactly is this suppose to be? For scalar multiplication or matrix-matrix multiplication... Also, the Multiplication Property of Negative One.. is that simply M * -1...

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the multiplicative-identity matrix is a matrix I where M * I = M. I'll leave this as an exercise to you to derive it.

As far as scalars go: cM = M would be the scalar multiplicative identity. Think about how scalar/matrix multiplication works and you can easily derive this one.

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Yes one of my questions is

Find matrix T where M * T = M
Find Matrix Z where M * Z = Z

I have been attempting to work out, I am going to walk myself through this thought that T = 1/M, the inverse of M.



As for scalar, I believe cM = M thus c = 1.0

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Quote:
Original post by Halsafar
Yes one of my questions is

Find matrix T where M * T = M
Find Matrix Z where M * Z = Z

I have been attempting to work out, I am going to walk myself through this thought that T = 1/M, the inverse of M.



As for scalar, I believe cM = M thus c = 1.0


Hi Halsafar,

it sounds like you are learning about vector spaces or group theory. One thing that you need to burn into your mind is that the inverse of multiplication is not assumed to be division. In particular, you should never think division when working with matrices, it is always multiplication by an inverse.


-Josh

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Yah -- I have beginning to see a lack of division anywhere... Always the inverse.

It seems my theory is incorrect tho --
In A * T = A, T = 1/A
A * 1/A != A

>.<

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Original post by Halsafar
In A * T = A, T = 1/A

Well, that's certainly incorrect. Try setting A = 3 and T = 1/3. BTW, a matrix's inverse should be expressed as M-1, not 1/M. Again, there is no division for matrices.

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Ok, lets look at this another way, M * P = MP (where MP = M)

So, doing this on a 2x2 we get:

M =
[a11, a12
a21, a22]
and
P =
[b11, b12
b21, b22]

MP =
[a11 * b11 + a12 * b21, a11 * b12 + a12 * b22
a21 * b11 + a22 * b21, a21 * b12 + a22 * b22]


Now with this we want:

a11 * b11 + a12 * b21 = a11
a11 * b12 + a12 * b22 = a12
a21 * b11 + a22 * b21 = a21
a21 * b12 + a22 * b22 = a22


Solve that and you can now generalize it for an nxn matrix.

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Quote:
Original post by jjd
Quote:
Original post by Halsafar
Yes one of my questions is

Find matrix T where M * T = M
Find Matrix Z where M * Z = Z

I have been attempting to work out, I am going to walk myself through this thought that T = 1/M, the inverse of M.



As for scalar, I believe cM = M thus c = 1.0


Hi Halsafar,

it sounds like you are learning about vector spaces or group theory. One thing that you need to burn into your mind is that the inverse of multiplication is not assumed to be division. In particular, you should never think division when working with matrices, it is always multiplication by an inverse.


-Josh



lol - tell that to MATLAB.

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Quote:
Original post by Halsafar
I'm sorry, I cannot find out what you mean...
I've working out trying to solve the last 4 but I'm lost.


It doesn't really matter. In real life, one practically never actually computes the inverse of a matrix - it's slow ( O(n^3), iirc), and unstable/inaccurate.

Rather, one thinks about what one wanted the inverse *for*, and deals with that issue directly. For example, if one wants to solve a linear system, algorithms like Cholesky, LU, and the like are relevant. If want wants to solve a least squares problem, QR, etc. becomes relevant.

I'd imagine that googling "numerical linear algebra" (with the quotes) would return an abundance of sources.

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