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brownman
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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.
brownman said:Okay the combined definitions from all of you seem to make a general sort of sense, thank you for the help guys :)
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.
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.
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.
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.
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.
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.