Inverse of a matrix + determinant

[JamesGoh]

To determine if a matrix is invertible or not, can we determine this by seeing if the determinant of the matrix is zero or non-zero ?

If it's zero, then the matrix doesn't exist because the inverse of the determinant would be an infinite number ?

 

[AlephZero]

To determine if a matrix is invertible or not, can we determine this by seeing if the determinant of the matrix is zero or non-zero ?

Yes.

If it's zero, then the matrix doesn't exist because the inverse of the determinant would be an infinite number ?

I think "matrix" was a typo for "inverse matrix".

You need to be careful about how you use the word "infinite" in mathematics, but your basic idea is right.

 

[HallsofIvy]

A reason for that is that the inverse of an invertible matrix is the matrix in which the "ij" term is the minor of the "ji" term of the original matrix divided by the determinant. Given an "ji" term, it has a minor so the only problem is that we cannot divide by 0.

 

[Bacle]

Another way of looking at it (for nxn-matrices):

Det(AB)=DetADetB;

In our case, say we have an inverse A' for A, then:

AA'=I , so that,

Det(AA')=DetADetA'=DetI=1,

So you need two numbers whose product equals one, and that rules out DetA=0.

 

[blue_raver22]

Yes, that is correct. The invertibility of a matrix is determined by its determinant. If the determinant is non-zero, then the matrix is invertible and its inverse can be calculated. However, if the determinant is zero, the matrix is not invertible and its inverse does not exist. This is because the inverse of a matrix is defined as the reciprocal of its determinant, and dividing by zero is undefined. Therefore, the determinant serves as a useful tool in determining the invertibility of a matrix.