If we have an n through n matrix referred to as A. Just how do we understand if over there is an inverse matrix A^-1 such that the product A * A^-1 is the n through n identity matrix?

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If the determinant the the matrix A (detA) is no zero, then this matrix has actually an inverse matrix. This building of a matrix deserve to be found in any type of textbook on greater algebra or in a textbook ~ above the theory of matrices.

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Yes Jose vegas is right. Infact, what girlfriend should have if det(A) is non-zero A.Inv(A) = Inv(A).A = i the identification matrix. However, for large value that n it is an overwhelming to discover det(A). If girlfriend apply, Gauss elimination method, then throughout elimintion procedure t some allude your diagonal facet becomes zero have the right to not be made non-zero by elementary heat exchange climate the matrix is singular and the station does no exist.
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If A is square you require det(A) nomzero. If A is no square and you don' t require the other condition A*A^(-1)=I, then you are in search of \"left inverses\", and also that is a lengthy story. Ns would inspect Wikipedia because that help. Cheers!

Yes Jose las vegas is right. Infact, what girlfriend should have if det(A) is non-zero A.Inv(A) = Inv(A).A = i the identification matrix. However, for huge value the n the is an overwhelming to uncover det(A). If you apply, Gauss removed method, then throughout elimintion procedure t some point your diagonal facet becomes zero can not it is in made non-zero by elementary heat exchange then the matrix is singular and also the inverse does not exist.
It relies on the matrix. If that is of type integer, then you have the right to do Gauss-Jordan elimination. If you don't finish up with a zero row, climate your procession is invertible. Of course computation that determinant for small n is more efficient. Other method is to try to find eigenvalues, if zero is not among them, then again A is invertible. Over there exist virtually ten different equivalent ways for your task.
How you pick to show existence of an inverse really depends on the matrix. There room instances wherein finding det(A) is far more difficult than proving .
In various other case, the product the the matrices A and, in this situation A^-1, will provide you a procession of a rank equal to the minimial location of the matrices A and A^-1. Therefore, you cannot acquire the identity matrix which have full rank equal n.

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I such way, even if the size of the procession is large, us may identify if the procession A is invertible without computation of its determinant.
R. Mittal's answer was the enough one in the 19th century (where nobody considered huge matrices). Today the just reasonable means is to make a singular worth decomposition (SVD) and also inspect the singular worths (i.e. Permit the computer do that). If at the very least one of lock is zero (I don't talk about the concern 'what is zero' for outcomes of an extensive numerical computations) then the matrix has actually no timeless inverse. It has a pseudo-inverse (Penrose inverse) though (which is closely related to the SVD), which deserve to replace the classical inverse in numerous applications whereby classically one used the inverse. Peter's contribution to that side the the problem is valuable. Of course, every this stuff is in Wikipedia, and also everybody who works with matrices in all but the most trivial applications need to know it!