Just wondering about the structure of a determinant

In summary, knowing the determinant of a matrix can tell you whether it is invertible or not, and it is the factor by which volumes are multiplied under the linear transformation represented by the matrix. However, it does not provide much information about the individual entries of the matrix or the cofactors across a specific row or column. In a complex vector space, the determinant is just the product of the eigenvalues and the trace of a matrix is the sum of the eigenvalues, which can be helpful in some cases.
  • #1
Dosmascerveza
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Just wondering about the "structure" of a determinant...

How much can a determinant tell you about the entries of a matrix? How much more if you know the size of the aforementioned matrix? How much more if you know that the matrix is symmetric?(perhaps a silly question). How much more if you know the cofactors across the ith row or jth column?
 
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  • #2


Very little. An n by n matrix contains [itex]n^2[/itex] entries and taking the determinant reduces all that information to one number. You will have lost an immense amount of information that you cannot get back just knowing the determinant.

The one thing you can be sure about, just by knowing the determinant of a matrix, is whether it is invertible or not.
 
  • #3


The determinant is the factor by which volumes are multiplied under the linear transformation that the matrix represents. That is all it tells you.
 
  • #4


In a complex vector space the determinant is just the product of the eigenvalues. Sometimes that is helpful, often times not. Likewise, the fact that the trace of a matrix is the sum of the eigenvalues is also sometimes of use.
 

FAQ: Just wondering about the structure of a determinant

What is a determinant?

A determinant is a mathematical concept used in linear algebra to determine the properties of a matrix. It is represented by a single number and is used to calculate the inverse of a matrix, solve systems of linear equations, and determine the area or volume of a geometric figure.

How is a determinant calculated?

The calculation of a determinant depends on the size of the matrix. For a 2x2 matrix, the determinant is calculated by multiplying the elements on the main diagonal and subtracting the product of the off-diagonal elements. For larger matrices, there are various methods such as the cofactor expansion method, row reduction, or using the Leibniz formula.

What is the significance of the determinant in linear algebra?

The determinant of a matrix is a useful tool in linear algebra as it helps determine if a matrix is invertible, which is essential in solving systems of equations. It also provides information about the linear independence of the columns or rows of a matrix and can be used to find the volume or area of a geometric figure.

What happens to the determinant when operations are performed on a matrix?

When a matrix undergoes elementary row operations (such as scaling, swapping, or adding rows), the determinant is affected in a predictable way. For example, scaling a row by a constant multiplies the determinant by that constant, while swapping two rows changes the sign of the determinant. These operations are useful when solving systems of equations using matrices.

Can the determinant be negative?

Yes, the determinant can be negative, positive, or zero. The sign of the determinant depends on the arrangement of the elements within the matrix. For example, a 2x2 matrix with a positive determinant means that the matrix is oriented clockwise, while a negative determinant means it is oriented counterclockwise. In general, a nonzero determinant means that the matrix is invertible, while a zero determinant means it is not invertible.

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