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 ?
Yes. 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.
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.
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.