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Math Help - 3 x 3 Matrix

  1. #1
    Fel
    Fel is offline
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    3 x 3 Matrix

    Consider the following formula for the determinant of a 3 x 3 matrix:

    det(x1y1z1//x2y2z2//x3y3z3) = x1y2z3+y1z2x3+z1x2y3 - (x3y2z1+y3z2x1+z3x2y1)

    a) Show that is it linear in the first column.
    b) Show that it is alternating when you switch the first and second column.

    (// denotes the next row in the matrix)

    I'm not sure how to prove this, any help is great. Thanks.
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  2. #2
    Super Member malaygoel's Avatar
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    Quote Originally Posted by Fel View Post
    Consider the following formula for the determinant of a 3 x 3 matrix:

    det(x1y1z1//x2y2z2//x3y3z3) = x1y2z3+y1z2x3+z1x2y3 - (x3y2z1+y3z2x1+z3x2y1)

    a) Show that is it linear in the first column.
    b) Show that it is alternating when you switch the first and second column.

    (// denotes the next row in the matrix)

    I'm not sure how to prove this, any help is great. Thanks.
    a) It says that if you multiply the first column by a number k, then the determinant also get multplied by k. To prove this, multiply x1,x2,x3 by k and see what happens to determinant.
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  3. #3
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    Quote Originally Posted by malaygoel View Post
    a) It says that if you multiply the first column by a number k, then the determinant also get multplied by k. To prove this, multiply x1,x2,x3 by k and see what happens to determinant.
    It also says that if you replace x1, x2, x3 by a1+ b1, a2+ b2, a3+ b3, you will get the determinant with a1, a2, a3 plus the determinant with b1, b2, b3.

    Tedious calculation but just basic algebra.
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