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I was wondering if anyone could think of a three-by-three determinant containing a and b (and other real numbers) whose expansion is ab(a + b)^2. There will probably be many possibilities but just one will do. Thanks.

Will
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2 Answers2

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If you mean

"there is a 3X3 matrix which contain a and b and other number. Determinant of the matrix = ab(a + b)^2"

Then my answer is (for a=1, b=2)

a    b    0
1   -2    6
1    3   -6

= 18 which is ab(a + b)^2

LORDTEK
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  • The determinant of the above is `12b-6a` which is clearly not what OP is looking for. On the other hand, it isn't 100% clear what OP had in mind. – John Coleman Jan 10 '16 at 13:02
  • 12b-6a use a=1 and b=2 = 18 = ab(a + b)^2 = 18 – LORDTEK Jan 10 '16 at 13:30
  • I know that for those particular values the formula works out but note that for just about any matrix containing `a` and `b` plus constant values the determinant will be an algebraic expression in `a` and `b` which can then be set equal to `ab(a+b)^2` and solved to find particular choices for `a` and `b` that work. What OP was clearly asking for was a matrix for which the determinant is equal to `ab(a+b)^2` for *all* `a` and `b`. By setting `a = 1, b = 2` you are adding constraints not present in the problem description. – John Coleman Jan 10 '16 at 13:37
  • I am not sure that he asked as your mind. Be patient and wait his response. – LORDTEK Jan 10 '16 at 15:35
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A quick solution is provided by a diagonal matrix:

a  0  0
0  b  0
0  0 (a+b)^2

This follows since the determinant of a diagonal matrix is the product of the numbers along the diagonal.

That is also true for triangular matrices, so for any choice of x,y,z, each of the following works:

a  x  y
0  b  z
0  0 (a+b)^2

a  0  0
x  b  0
y  z (a+b)^2

This shows that (not surprisingly) there are infinitely many solutions. If you want a non-triangular solution, let your matrix be the product of two matrices of the form

a  x  y
0  b  z
0  0  1

and

1   0   0
w  a+b  0
u   v  a+b

this will work since det(AB) = det(A)*det(B) and the determinant of the first matrix is ab while the determinant of the second matrix is (a+b)^2

John Coleman
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