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.
Best Answer
In general, one can reduce the matrix to upper triangular form using Gauss elimination which has a cost of O(n^3). From there, the determinant is proportional to the product of the diagonal entries of this matrix. So if one of the diagonal entries is zero, determinant will be zero.