Why this property of the product of two matrices

This means that for any matrix C that satisfies PB=0, we can obtain another matrix D=BC that also satisfies PB=0.Furthermore, the property det(PPt) = det(BtB) = 7778 is not a coincidence either. It is a result of the fact that det(AB) = det(A)det(B) and the fact that det(Pt) = det(P). So, we can choose any matrix C such that det(C)=sqrt(7778) and obtain a new B such that det(PPt) = det(BtB) = 7778. In summary, the property det(PPt) = det(BtB) = 7778 means that there exists an
  • #1
senmeis
69
2
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(PPt) = det(BtB) = 7778

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

Senmeis
 
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  • #2
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(PPt) = det(BtB)?
 
  • #3
Pb=0
 
  • #4
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(DTD)=det(BTB)=7778.

See if you can take this further to ensure that all elements of D are integers.
 
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  • #5
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
 
  • #6
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.
 

FAQ: Why this property of the product of two matrices

1. Why is the product of two matrices not commutative?

The product of two matrices is not commutative because the order in which matrix multiplication is performed matters. This means that AB is not equal to BA in most cases. In mathematical terms, matrix multiplication does not follow the commutative property, which states that changing the order of operands does not change the result.

2. Why does the product of two matrices result in a new matrix?

The product of two matrices results in a new matrix because it combines the information from both matrices in a specific way. Each element in the resulting matrix is calculated by multiplying a row from the first matrix by a column from the second matrix and adding the products. This process results in a new matrix with different dimensions and values than the original matrices.

3. Why does the number of columns in the first matrix have to equal the number of rows in the second matrix for multiplication to be possible?

The number of columns in the first matrix and the number of rows in the second matrix have to be equal for multiplication to be possible because of the rules of matrix multiplication. In order for the multiplication to be valid, the inner dimensions (columns in the first matrix, rows in the second matrix) need to match. This ensures that the dimensions of the resulting matrix are defined and that the multiplication is carried out correctly.

4. Why does the product of two matrices result in a different matrix size than the original matrices?

The product of two matrices results in a different matrix size because the number of rows and columns in the resulting matrix is determined by the number of rows and columns in the original matrices. The number of rows in the resulting matrix is equal to the number of rows in the first matrix, and the number of columns in the resulting matrix is equal to the number of columns in the second matrix.

5. Why is matrix multiplication used in various scientific and mathematical applications?

Matrix multiplication is used in various scientific and mathematical applications because it provides a powerful and efficient way to represent and manipulate data. It allows for complex calculations and transformations to be performed on large sets of data, making it useful in fields such as physics, engineering, economics, and computer science.

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