Determinant multiplication

In summary: It is true that the multiplication of determinants is commutative. This means that the order in which we multiply two determinants does not affect the result, and thus |A||B| = |B||A| = |AB| = |BA|. This implies that all is right with the world! Additionally, the determinant has multiple uses, such as measuring linear measures and testing linear dependence of vectors. In the context of matrix multiplication, the determinant also provides information on how a transformation changes another system, and the commutative property is useful in preserving length in compositions of maps. This is relevant in physics, where certain groups have properties similar to rotation matrices.
  • #1
131
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is multiplication of determinant commutative ?
if so , then is this correct: |A||B|=|B||A|=|AB|=|BA| ?
then what does |AB|=|BA| imply ??
 
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  • #2
hi phymatter! :wink:
phymatter said:
is multiplication of determinant commutative ?
if so , then is this correct: |A||B|=|B||A|=|AB|=|BA| ?

yup! :smile:
then what does |AB|=|BA| imply ??

that all is right with the world! :wink:
 
  • #3
phymatter said:
is multiplication of determinant commutative ?
if so , then is this correct: |A||B|=|B||A|=|AB|=|BA| ?
then what does |AB|=|BA| imply ??

The determinant has a lot of different uses.

In one sense it measures a linear measure in some dimension (in general we call it a parallelpiped in dimension n).

Another common use is to test linear dependence of a set of vectors (unit basis or elsewise).

Typically when we think about the product of two matrices, one way to view them is to view them as a composition of maps (or composition of linear functions).

Like if we had a function f(g(x)) using a AB we have f(g(A)) where A isn't necessarily a single number or even a vector, but instead a general matrix.

So this measure of a transformation gives information to how it changes another system (usually a vector but sometimes a matrix).

For example a rotation matrix has a determinant of 1. This means that the transformation preserves length when applied to some vector input. So if you had two rotation matrices and you used composition of maps (in this case ABC where they are all rotation matrices), then the composition of all these maps applied to any vector will conserve the length and computationally this can be proved using det(ABC) = det(A) x det(B) x det(C) = 1 x 1 x 1 = 1. This is a good property to have since any composition of rotations will preserve length.

In physics, you will have things called groups and a lot of groups used in high level physics have certain properties like the rotation matrices I mentioned above. Things like Special Unitary (SU(N)) have this kind of property.
 
  • #4
thanks tiny-tim and chiro!
 
  • #5


Determinant multiplication is a mathematical operation used to calculate the determinant of a matrix. It involves multiplying the elements of a matrix and taking the sum of their products in a specific way. The result is a single number that represents the scaling factor of the matrix.

To answer the question, yes, the multiplication of determinants is commutative. This means that the order in which the determinants are multiplied does not affect the final result. This can be represented by the equation |A||B|=|B||A|, which shows that the determinant of matrix A multiplied by the determinant of matrix B is equal to the determinant of matrix B multiplied by the determinant of matrix A.

Furthermore, |AB|=|BA| implies that the determinant of the product of two matrices is equal to the determinant of the product of the matrices in reverse order. In other words, the order in which the matrices are multiplied does not affect the determinant of the final product. This property is important in solving systems of linear equations and in finding the inverse of a matrix.

In conclusion, determinant multiplication is commutative, and this property allows for more efficient and flexible calculations in linear algebra.
 

1. What is a determinant?

A determinant is a mathematical value that can be calculated from a square matrix. It is used to determine certain properties of the matrix, such as whether it is invertible or singular.

2. How is the determinant of a matrix calculated?

The determinant of a matrix is calculated by using a specific formula. For a 2x2 matrix, the determinant is found by multiplying the top left element by the bottom right element and subtracting the product of the top right and bottom left elements. For larger matrices, more complex formulas are used.

3. What is the significance of determinant multiplication?

Determinant multiplication is used to find the determinant of a matrix that has been multiplied by a scalar value. This can help determine if the matrix is invertible, and can also be used to solve systems of linear equations.

4. Can a determinant be negative?

Yes, a determinant can be negative. This occurs when the matrix has an odd number of negative eigenvalues. However, the negative sign does not affect the overall value of the determinant, which is still a measure of the matrix's size.

5. What are some real-world applications of determinant multiplication?

Determinant multiplication is used in a variety of fields, such as physics, engineering, and economics. It is used to solve systems of linear equations, determine the stability of a system, and calculate the area of a parallelogram or the volume of a parallelepiped.

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