Why is a matrix singular if the determinant is zero?

In summary, the determinant of an alternating multilinear map is the product of the map's rows (or columns).
  • #1
brownman
13
0
I'm looking for the deeper meaning behind this law/theorem/statement (I don't know what it is, please correct me). My textbook just told us a matrix is not invertible if the determinant is zero.
 
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  • #2
hi brownman! :smile:

an nxn matrix is a linear transformation on ℝn to itself

its determinant is the ratio of the n-dimensional volume of the image of a box to the volume of the box itself

if the determinant is 0, the image of the box is flattened by at least one dimension (because its volume is 0), so the transformation obviously isn't invertible (it isn't one-to-one)

if the determinant isn't 0, the box isn't flattened, so the transformation can always be reversed :wink:
 
  • #3
It is also true that det(AB)= det(A)det(B). so if det(A)= 0, it is impossible to have AB= I for any matrix B.
 
  • #4
As the determinant is the product of the eigenvalues of a matrix it being zero means at least one of the eigenvalues is zero as well. By definition it follows that Ax = 0x = 0 for some vector x ≠ 0. In case A was invertible we would have (A^-1)Ax = 0 meaning x = 0 which contradicts that x ≠ 0 and therefore A is not invertible.
 
  • #5
Okay the combined definitions from all of you seem to make a general sort of sense, thank you for the help guys :)
 
  • #6
A definition of the determinant of an n*n matrix as the n-volume spanned by its column vectors gives this result easily. It can also be proved using other definitions with somewhat more hassle.
 
  • #7
brownman said:
Okay the combined definitions from all of you seem to make a general sort of sense, thank you for the help guys :)

A way to "see" that a determinant is a volume measure is to work it out for two vectors in the plane. It is a simple exercise in Euclidean geometry.

For instance suppose the two vectors are (5,0) and (3,2)

The area of the parallelogram that they span is the height times the base which is 2(the height) x 5(the base). The determinant of the matrix

5 0
3 2

is 5 x 2 - 3x0.

Can you generalize this example?
 
  • #8
fortissimo said:
A definition of the determinant of an n*n matrix as the n-volume spanned by its column vectors gives this result easily. It can also be proved using other definitions with somewhat more hassle.

Also, it is the signed volume, which depends on the order of the vectors.
 
  • #9
Another way to think about determinates is to think of them as multilinear functions of the rows (or columns) and investigate how this algebraic property is related to computing volumes. The approach leads to the idea of the exterior algebra of a vector space.

Question: Is every alternating multilinear map a determinant?
 

1. Why is a matrix singular if the determinant is zero?

The determinant of a matrix is a value that represents the scaling factor of the transformation that the matrix represents. If the determinant is zero, it means that the transformation is not one-to-one, meaning that there exists more than one input vector that maps to the same output vector. This leads to a loss of information, making the matrix singular.

2. Can a matrix be singular if the determinant is not zero?

No, a matrix can only be singular if the determinant is zero. This is because a non-zero determinant indicates that the matrix is invertible, meaning that it has a unique inverse and can preserve information. A zero determinant, on the other hand, means that the matrix is not invertible and cannot preserve all information.

3. What is the significance of a singular matrix?

A singular matrix has no inverse, meaning that it cannot be used to solve linear equations. This can be a problem in various applications that involve solving systems of equations, such as in engineering or physics. A singular matrix also cannot represent a unique transformation, which can affect the accuracy of calculations and predictions.

4. How can we determine if a matrix is singular without calculating the determinant?

There are other methods for determining if a matrix is singular without calculating the determinant. One way is to check if the rank of the matrix is less than its dimensions. If the rank is less, it means that there are linearly dependent rows or columns in the matrix, which leads to a zero determinant. Another method is to check for zero eigenvalues using the eigenvalue decomposition or singular value decomposition.

5. Can a singular matrix be used for any calculations?

While a singular matrix cannot be used for solving equations or representing unique transformations, it can still be useful in certain applications. For example, in data compression, a singular matrix can represent a lossy compression technique that reduces the size of data by removing redundant information. Singular matrices can also be used in other areas such as computer graphics and cryptography.

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