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Math Help - Plugging Matrixes into Functions

  1. #1
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    Plugging Matrixes into Functions

    I'm taking linear algebra and I'm not really understanding how putting a matrix into a function works. We learned that f(x) = A^2 + 2A + 1 means square matrix A + 2* matrix A + Identity Matrix. Our teacher said you can basically plug anything into a function. But does the function still obey the laws of algebra?

    For example can you factor a function? I have a question that says show p1(A) = p2(A)p3(A) for any square matrix A. It tells me p1(x) = x^2 + 9 and p2(x) = x + 3 and p3(x) = x - 3. I verified it for a specific 2x2 matrix A, but am not sure how to generalize. I would like to generalize it even farther than they are asking to factor anything like I can with regular algebra.

    I have a similar question later, show that a square matrix A satisfies A^2 - 3A + I = 0 then A^-1 = 3I - A. It is trivial to show it works for a specific matrix A, and I would like to understand how they came up with the 2nd equation, and not merely that it is true, so I can understand what operations are valid.

    I don't really know the name of what I'm having trouble with, so I wasn't really able to find anything useful on the internet. I'm taking it at a community college, so the people in the math lab can't really help me, in fact I work in our math lab myself.
    Thanks so much for your help!
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  2. #2
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    Quote Originally Posted by jjoshua2 View Post
    I'm taking linear algebra and I'm not really understanding how putting a matrix into a function works. We learned that f(x) = A^2 + 2A + 1 means square matrix A + 2* matrix A + Identity Matrix. Our teacher said you can basically plug anything into a function. But does the function still obey the laws of algebra?
    Well, you can plug anything you can add and multiply and multiply by numbers into a polynomial. You can even put them into more complicated functions if you can write the functions as infinite power series.

    For example can you factor a function? I have a question that says show p1(A) = p2(A)p3(A) for any square matrix A. It tells me p1(x) = x^2 + 9 and p2(x) = x + 3 and p3(x) = x - 3.
    I hope it didn't tell you that! (x+3)(x-3)= x^2- 9, not x^2+ 9! Was that what you meant?

    I verified it for a specific 2x2 matrix A, but am not sure how to generalize. I would like to generalize it even farther than they are asking to factor anything like I can with regular algebra.
    Just be careful about "commutativity". Numbers commute under multiplication (ab= ba) while matrices do not (in general AB\ne BA).
    (A+B)(A+B)= A^2+ AB+ BA+ B^2 but that is NOT A^2+ 2AB+ B^2 as it would be for numbers.
    Matrices are just like numbers except that multiplication is not commutative and not every matrix has a multiplicative inverse. For example if you know AB= 0, for matrices A, B, and the zero matrix, you cannot conclude that A= 0 or B= 0.

    I have a similar question later, show that a square matrix A satisfies A^2 - 3A + I = 0 then A^-1 = 3I - A. It is trivial to show it works for a specific matrix A, and I would like to understand how they came up with the 2nd equation, and not merely that it is true, so I can understand what operations are valid.
    A(A-3)= (A-3)A= -I so that A(3-A)= (3-A)A= I. By definition, A^{-1} is the vector such that A^{-1}A= AA^{-1}= I.

    I don't really know the name of what I'm having trouble with, so I wasn't really able to find anything useful on the internet. I'm taking it at a community college, so the people in the math lab can't really help me, in fact I work in our math lab myself.
    Thanks so much for your help!
    Follow Math Help Forum on Facebook and Google+

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