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Math Help - How to find the determinant of matrix if given characteristic equation only?

  1. #1
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    How to find the determinant of matrix if given characteristic equation only?

    hey there..

    The question is..
    lambda = x (i dont know how to type the symbol..)

    Let A be n x n square matrix. Suppose that
    det(xI-A) = c(x) = x^3- 3x^2 + x + 6..

    I know the answer is -6 by substitute lambda@x with 0..
    However.. i just want to know.. why should i substitute it with 0..?

    Thanks in advance..
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  2. #2
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    Re: How to find the determinant of matrix if given characteristic equation only?

    Quote Originally Posted by nameck View Post
    hey there..

    The question is..
    lambda = x (i dont know how to type the symbol..)

    Let A be n x n square matrix. Suppose that
    det(xI-A) = c(x) = x^3- 3x^2 + x + 6..

    I know the answer is -6 by substitute lambda@x with 0..
    However.. i just want to know.. why should i substitute it with 0..?

    Thanks in advance..
    You must be careful.

    c(x)=\text{det}(xI-A)=x^3-3x^2+x+6

    If you use x=0 you get

    c(0)=\text{det}(-A)=6 \implies (-1)^3\text{det}(A)=6
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  3. #3
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    Re: How to find the determinant of matrix if given characteristic equation only?

    yup2... so.. -(det A) = 6.. so.. det A = -6 right?
    ok then.. why it is 0?

    Got one more question..
    if i've got 4 x 4 matrix,
    the sign will be positive?
    (-1)^4 det(A) correct?
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  4. #4
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    Re: How to find the determinant of matrix if given characteristic equation only?

    Quote Originally Posted by nameck View Post
    hey there..

    The question is..
    lambda = x (i dont know how to type the symbol..)

    Let A be n x n square matrix. Suppose that
    det(xI-A) = c(x) = x^3- 3x^2 + x + 6..

    I know the answer is -6 by substitute lambda@x with 0..
    However.. i just want to know.. why should i substitute it with 0..?

    Thanks in advance..
    Since the det(xI- A), with x= 0 is just det(0-A). Multiplying one row of a matrix by -1 multiplies the determinant -1. Multiplying A by -1 multiplies all three rows by -1 and so multiplies the determinant by -1.
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  5. #5
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    Re: How to find the determinant of matrix if given characteristic equation only?

    Yes, if you multiply a 4 by 4 matrix by -1, you have multiplied all 4 rows by -1 and so multiply the determinant by (-1)^4= 1
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  6. #6
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    Re: How to find the determinant of matrix if given characteristic equation only?

    Thanks HallsofIvy.. U are always answering my question perfectly..
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