Why this property of the product of two matrices

[senmeis]

Hello,

The product of a 2x5 matrix P and a 5x3 matrix B shall be a 2x3 zero matrix. P and B are all matrices of integers.

P = [6 2 -5 -6 1;3 6 1 -6 -5]

One possible B is [0 -4 0;3 0 0;-1 -1 3;2 -3 -2;1 1 3]

This solution B has a property: det(PP^t) = det(B^tB) = 7778

The question is: What does this property mean? How to get another B with this property?

Senmeis

 

[HallsofIvy]

I really don't know how you got B or why you refer to it as a "solution". Solution to what problem? Just having the property that det(PP^t) = det(B^tB)?

 

[senmeis]

Pb=0

 

[D H]

Post-multiply B by any 3x3 matrix C, i.e., D=BC, and you'll have another 5x3 matrix D that satisfies PD=0. If C is unimodular (determinant = ±1), then det(D)=±det(B), so det(D^TD)=det(B^TB)=7778.

See if you can take this further to ensure that all elements of D are integers.

 

[senmeis]

Yes, you are right, but I still don’t know if this property is by chance. More important, I can’t even get another B that fulfills this property.

Senmeis

 

[D H]

No, it is not by chance. Matrix multiplication is associative: (PB)C = P(BC). If PB=0 then P(BC) must necessarily be zero as well.