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Thread: Find Determinant of the Matrix

  1. #1
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    Find Determinant of the Matrix

    Hi! My HW task is to find determinant of given matrix. I tried to do elementary row operations, but did not get anything. How to get rid of ?
    Answer should be 0 for p>2.

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  2. #2
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    Re: Find Determinant of the Matrix

    I don't think it's necessary to get rid of B.

    I would try to apply the definition of determinant directly and use the principle of induction to show the determinant is 0 for p>2.
    The key, I think, is using the inductive step when asked to calculate the determinant of minors for matricies larger the 2x2.
    Last edited by MacstersUndead; May 3rd 2018 at 06:36 AM.
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  3. #3
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    Re: Find Determinant of the Matrix

    After looking at the problem more, I think the problem can be solved using row reduction, and not induction like I originally suspected. I would consider looking at the case for p=3 first and observe the powers of a. Row reduce. I'll leave it to you to generalize. Process should be similar for greater than 3

    Original Matrix for p = 3

    a^n - B & a^n+1 - B & a^n+2 - B
    a^n+3 - B & a^n+4 - B & a^n+5 -B
    a^n+6 - B & a^n+7 - B & a^n+8 - B

    Row 2 minus Row 1 and Row 3 minus Row 2
    a^n - B & a^n+1 - B & a^n+2 - B
    a^n * (a^3 - 1) & a^n+1 * (a^3-1) & a^n+2 * (a^3-1)
    a^(n+3)* (a^3 - 1) & a^n+4 * (a^3 -1) & a^n+5 (a^3 - 1)

    Now note that Row 3 is a^3 * Row 2. Hence the last row can be reduced to 0, hence proving the determinant is 0.
    Last edited by MacstersUndead; May 3rd 2018 at 12:48 PM.
    Thanks from Vinod
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