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Homework Help: Matrix question

  1. Jul 6, 2010 #1
    1. The problem statement, all variables and given/known data
    Use the matrix capabilities of a graphing utility to find:
    [tex]f(A)=a_{0}I_{n}+a_{1}A+a_{2}A^2+\cdots+a_{n}A^n[/tex]

    1.
    [tex]f(x)=x^2-5x+2[/tex]
    [tex]A=\left[\begin{array}{cc}2&0\\4&5\end{array}\right][/tex]

    2. Relevant equations



    3. The attempt at a solution

    Well, I know the answer is
    [tex]\left[\begin{array}{cc}-4&0\\8&2\end{array}\right][/tex]
    However, I don't know how to get it.

    I would think you would do A^2-5A+2 however you cant add a constant to a matrix. I'm not sure exactly what I'm supposed to do.
     
  2. jcsd
  3. Jul 6, 2010 #2
    The constant 2 might be
    [tex]
    \left[\begin{array}{cc}2&2\\2&2\end{array}\right]
    [/tex]
     
  4. Jul 6, 2010 #3
    Nope, thats not it. Just tried it and it's wrong, not sure how it would be it though.
     
  5. Jul 6, 2010 #4
    A constant by itself might signify that it is in operation with an identity. So 2 might be
    [tex]

    \left[\begin{array}{cc}2&0\\0&2\end{array}\right]

    [/tex]
     
  6. Jul 6, 2010 #5
    oh, ok. That's turns out to be right. Thanks!
     
  7. Jul 6, 2010 #6

    Mark44

    Staff: Mentor

    Right. The polynomial is f(A) = A2 - 5A + 2I.
     
  8. Jul 6, 2010 #7
    How exactly did you derive 'I's value?
     
  9. Jul 6, 2010 #8

    Mark44

    Staff: Mentor

    I is the 2 x 2 identity matrix, defined as
    [tex]\left[\begin{array}{cc}1&0\\0&1\end{array}\right][/tex]

    Since A is given as a 2 x 2 matrix, the appropriate identity matrix must also be 2 x 2. If A were given as a 3 x 3 matrix, you would need to use the 3 x 3 identity matrix, which is defined as
    [tex]\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right][/tex]

    The form of the identity matrix to use depends on the size of the square matrices being used in the problem.
     
  10. Jul 6, 2010 #9
    Ahh.. ok. I didn't know what an identity matrix was before. But now I know. Interesting.
     
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