Q. What is the main condition for the existence of the inverse of a matrix?
Context: I am an Engineering Student. Currently I am in 1st year of college. When I was in class XII, I learnt how to find the inverse of a matrix:
If $A$ is a $n×n$ matrix, then its inverse is $$A^{-1}=\dfrac{1}{|A|}(\mathrm{adj}(A))$$.
From this formula, it is clear that a necessary condition for existence of inverse of a matrix is $|A|\neq0$.
Till class XIIth, I learnt only about $3×3$ matrices. So, finding the determinant of a $3×3$ matrix is quite easy. But if we are given a $8×8$ matrix, then how to find the determinant of that matrix? For example- If
$A$ = $$ \begin{bmatrix} 2 & 4 & 1 & 7 & 9 & 4 & 11 & 8 \\ 3 & 3 & 6 & 1 & 5 & 2 & 14 & 3 \\ 6 & 7 & 8 & 9 & 1 & 4 & 22 & 2 \\ 1 & 4 & 7 & 2 & 2 & 9 & 18 & 7 \\ 5 & 5 & 5 & 5 & 9 & 9 & 27 & 9 \\ 6 & 7 & 1 & 7 & 3 & 8 & 29 & 6 \\ 7 & 4 & 2 & 2 & 2 & 5 & 33 & 12 \\ 14 & 5 & 6 & 5 & 6 & 7 & 36 & 15 \\ \end{bmatrix} $$
then finding the determinant is a very difficult for such a matrix.
I want to know what is the main condition for the inverse of a matrix to exist ? In many books as well, I find that the authors have mentioned that if the determinant of a matrix is not equal to $0$ then the inverse exists. Finding determinants is easy for $3×3$ and $4×4$ matrices, but for $n\geq5$ finding the determinant becomes difficult. \
Please let me know about any other method(s) to check whether the inverse of a matrix exists.
As noticed, a square matrix is invertible if and only if its columns (and rows) are linearly independent.
The idea is that under this condition, since the columns (and rows) of $A$ are a basis, the system $Ax=b$ has exactly one solution for any $b$, therefore if we indicate with $e_i$ the $i^{th}$ basis vector for the canonical basis, the system $Ax=e_i$ has exactly one solution $v_i$. Therefore, if we consider the matrix $B$ which has the vectors $v_1,v_2,\ldots,v_n$ as columns we have $AB=I$. Matrix $B$ is therefore unique and is named the inverse matrix $A^{-1}$ of $A$.
As noticed, for not small matrices, in general Gauss elimination is the most effective method to determine whether the inverse exists, that is whether or not columns (and rows) are lineraly independent.
Note also that with some more effort by Gauss-Jordan method we can also calculate the inverse matrix. The idea is start from the extended matrix $[\;A\;|\:I\;]$ and perform rows operation on the entire matrix in order to obtain the identity matrix on the left part, at the end of the process we'll find the inverse matrix on the right part, that is $[\;I\;|\:A^{-1}\;]$. The reason is that rows operation are equivalent to multiplication from the left by a matrix such that $BA=I$, that is $B=A^{-1}$ and then
$$A^{-1}[\;A\;|\:I\;]=[\;A^{-1}A\;|\:A^{-1}I\;]=[\;I\;|\:A^{-1}\;]$$