If the matrix $A$ is diagonalizable, then we know that its similar diagonal matrix $D$ has determinant $0$, so the matrix $A$ itself is invertible? However, if $A$ not diagonalizable, how are we sure that the matrix $A$ which has $0$ as eigenvalue is not invertible?
Here I have another confusion, does the degree of the characteristic polynomial determine the size of matrix. i.e. $\lambda (\lambda+2)^3 (\lambda-1)^2$ has $6\times 6$ matrix?
Best Answer
The determinant of a matrix is the product of its eigenvalues. So, if one of the eigenvalues is $0$, then the determinant of the matrix is also $0$. Hence it is not invertible.