I know that if $A$ is a square matrix of order $n$ , then :
$$A*\mathbb{adj}(A)=|A|I_n\cdots(1)$$
where $I_n$ is identity matrix of order $n$.
By definition of inverse of matrix ( which is $AA^{-1}=I$) , and also the fact that inverse must be unique , I can deduce from equation $1$ that :
$$A^{-1}= \frac{\mathbb{adj}(A)}{|A|}$$
given that $|A| ≠0$
This is because, it's valid to divide both side by a non-zero number , here which is $|A|$ .
Now , if $|A|=0$ , we cannot divide both side by zero , hence the only thing I get is :
$$A*\mathbb{adj}(A)={O}$$
But I cannot deduce from this equation that why an inverse can not exist. My teacher said that it's because if determinant is zero , then we cannot divide both side by zero , hence we can't get our inverse. But how can I ensure that there isn't any other method or equation through which I can obtain the inverse ? In other words, how can I ensure that the equation $1$ above in this post is the only way to obtain an inverse , so that If I can't get the inverse from equation $1$ , then I can't get it from anywhere else ? So my question is :
How to prove that a singular matrix have no inverse?
Best Answer
For square matrices, it holds that $\det AB = \det A \cdot \det B$. If $A$ was singular and had an inverse $A^{-1}$, then $0=\det A\det A^{-1}=\det E = 1$.
More generally, if one of the terms is singular, then the product is also singular.