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fox1
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I need to prove that a 3x3 matrix with all odd entries will have a determinant that is a multiple of 4.
This is how I set it up:
I let A = { {a, b, c}, {d, e, f}, {g, h, i} } with all odd entries
then I define B = { {a, b, c}, {d + na, e + nb, f + nc}, {g + ma, h + bm, i + cm} }
where I add the multiple of first row to second and third row. So only the first row will have odd integers entries while the second and third row will be even entries.
det(A) = det(B) since adding multiple of one row to another doesn't change the determinant
After this I was going to show that the each of three 2x2 matrix will have a even determinants. This is where I'm kind of stuck. I can show that the determinant of 3x3 will be even, but how can I show that it will be a multiple of 4?
This is how I set it up:
I let A = { {a, b, c}, {d, e, f}, {g, h, i} } with all odd entries
then I define B = { {a, b, c}, {d + na, e + nb, f + nc}, {g + ma, h + bm, i + cm} }
where I add the multiple of first row to second and third row. So only the first row will have odd integers entries while the second and third row will be even entries.
det(A) = det(B) since adding multiple of one row to another doesn't change the determinant
After this I was going to show that the each of three 2x2 matrix will have a even determinants. This is where I'm kind of stuck. I can show that the determinant of 3x3 will be even, but how can I show that it will be a multiple of 4?
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