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Thread: matrices

  1. #1
    Super Member bigwave's Avatar
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    Unhappy matrices

    $\textsf{If A and b are $n \times n$ matrices. Mark each statemant T or F. Justify each answer}\\$
    $\textit{ a. If $Ax=\lambda x$ for some vector x, then $\lambda$ is an eigenvalue of $A$}\\$
    $\textit{ b. a matrix A is not invertible if and only if 0 is an eigenvalue of A}\\$
    $\textit{ c. A number c is an eigenvalue of A iff the equation
    $(A-c)x=0$ has a nontrival solution}\\$
    $\textit{ d. Finding an eigenvector of A may be difficult
    but checking whether a given vector is in fact an eigenvector is easy}\\$
    $\textit{ e. To find the eigenvalue of A, reduce A to echelon form}$

    ok barely had time to post this
    but was having ???? with this

    deeply appreciate help
    having hard time in class
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  2. #2
    MHF Contributor

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    Re: matrices

    Quote Originally Posted by bigwave View Post
    $\textsf{If A and b are $n \times n$ matrices. Mark each statemant T or F. Justify each answer}\\$
    $\textit{ a. If $Ax=\lambda x$ for some vector x, then $\lambda$ is an eigenvalue of $A$}\\$
    If x= 0, that is true for any A and any $\lambda$.

    $\textit{ b. a matrix A is not invertible if and only if 0 is an eigenvalue of A}\\$
    A matrix A is not invertible if and only if its determinant is 0. What does that tell you about its eigenvalues?

    $\textit{ c. A number c is an eigenvalue of A iff the equation
    $(A-c)x=0$ has a nontrival solution}\\$
    That is equivalent to saying that Ax= cx for some non-zero x. Look back at the first statement above. What is the DEFINITION of "eigenvalue"?
    $\textit{ d. Finding an eigenvector of A may be difficult
    but checking whether a given vector is in fact an eigenvector is easy}\\$
    Finding an eigenvalue is equivalent to solving the eigenvalue equation, an n-degree polynomial for an n by n matrix. Compare solving, say, a fifth degree equation with checking whether or not a number is a solution by putting it into the equation.

    $\textit{ e. To find the eigenvalue of A, reduce A to echelon form}$
    The eigenvalues of A and the eigenvalues of the echelon form of A are NOT, in general, the same.

    ok barely had time to post this
    Did you have time to try to do it yourself?

    but was having ???? with this

    deeply appreciate help
    having hard time in class
    Perhaps it would help to focus on the basic definitions.
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  3. #3
    MHF Contributor
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    Re: matrices

    Quote Originally Posted by bigwave View Post
    $\textsf{If A and b are $n \times n$ matrices. Mark each statemant T or F. Justify each answer}\\$
    $\textit{ a. If $Ax=\lambda x$ for some vector x, then $\lambda$ is an eigenvalue of $A$}\\$
    $\textit{ b. a matrix A is not invertible if and only if 0 is an eigenvalue of A}\\$
    $\textit{ c. A number c is an eigenvalue of A iff the equation
    $(A-c)x=0$ has a nontrival solution}\\$
    $\textit{ d. Finding an eigenvector of A may be difficult
    but checking whether a given vector is in fact an eigenvector is easy}\\$
    $\textit{ e. To find the eigenvalue of A, reduce A to echelon form}$

    ok barely had time to post this
    but was having ???? with this

    deeply appreciate help
    having hard time in class
    a) F: $A0=\lambda 0$ for any $A,\lambda$.
    b) T
    c) Depends on what notation you consider well-defined. Is a matrix plus a scalar well defined? Or should it be $(A-cI)x = 0$?
    d) T
    e) F: $\begin{pmatrix}1 & -1 \\ 1 & 3\end{pmatrix}$ has eigenvalues 2,2. $\begin{pmatrix}1 & -1 \\ 0 & 4\end{pmatrix}$ has eigenvalues 1,4.
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  4. #4
    Super Member bigwave's Avatar
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    Re: matrices

    ok much mahalo
    we are on break so trying to a handle on this before 03/2

    the forum is much more helpfull than the book
    have 5 more probs
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