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rvc

  • one year ago

Matrices. Please help :) For what value of x will the following matrix A be of rank 1) equal to 3 2) less than 3

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  1. anonymous
    • one year ago
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    where is the matrix?

  2. rvc
    • one year ago
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    \[A=\left[\begin{matrix}3-x & 2 & 2 \\ 1 &4-x & 0 \\ -2 & -4 & 1-x\end{matrix}\right]\]

  3. anonymous
    • one year ago
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    really, you need to find x so that the determinant is non-zero which will make the matrix have full rank (3). and then find x so that the determinant is 0 then the matrix will not have full rank which means the rank will be less than 3

  4. rvc
    • one year ago
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    how do i do that

  5. anonymous
    • one year ago
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    do you know how to calculate the determinant?

  6. rvc
    • one year ago
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    should i equate the determinant to zero?

  7. anonymous
    • one year ago
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    once you get the expression for the determinant, you should have a 3rd degree polynomial. if you find the roots of that polynomial, then when x is the value of anoy of the roots, the determinant will be 0 and the second part of your question will be answered. if you pick any value that is not a root, then the determinant will be non-zero and you will have answered the first part of the question.

  8. rvc
    • one year ago
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    oh okay-tysm

  9. anonymous
    • one year ago
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    do you understand? the dots i'm trying to help you connect?

  10. rvc
    • one year ago
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    yep

  11. anonymous
    • one year ago
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    the first part is actually the easier of the two because you can use trial and error and there are an infinite number of values that will work. but for part 2, a maximum of 3 values work, and only one value if the polynomial has a pair of complex roots!

  12. anonymous
    • one year ago
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    if you are doing by hand, you can use the rational roots theorem to find the roots.

  13. anonymous
    • one year ago
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    did you find the answers?

  14. rvc
    • one year ago
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    nope

  15. anonymous
    • one year ago
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    did you get the polynomial?

  16. anonymous
    • one year ago
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    you there?

  17. anonymous
    • one year ago
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    did you want help finding the answers?

  18. anonymous
    • one year ago
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    the rank is 3 when the null space is trivial, i.e. only zero; this means that the entire codomain is mapped to. so we want \(\det A\ne 0\).

  19. anonymous
    • one year ago
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    if the null space is not trivial, then we will invariably have incomplete rank and so for rank less than 3 we want to solve \(\det A=0\)

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